In the first part of my answer I discuss whether a tidally locked planet can have life. In the other four parts I discuss various ways to get a bright night sky.
Part One of Five: Habitability of a tidally locked planet.
If you are asking about the permanent night of the eternally dark side of a tidally locked planet, then you have to worry about whether it will be warm enough for life or a frozen, lifeless, wasteland.
One problem with having habitable planets orbiting dim stars, the majority of stars in the universe, is that the habitable zones of those stars will be so close to those stars that the planets will be tidally locked. A minor change in the mass of a star will cause a much larger change in its total luminosity. Thus reducing the mass of a star slightly will reduce the size of its habitable zone much more, and thus a planet in the habitable zone will experience much more intense gravity from its star.
If the star is dim enough, the tidal forces from that star will tidally lock the planet so that one side will always face away from the star and the other side will always face the star.
At the close orbital distances, which planets around red dwarf stars would have to maintain for liquid water to exist at their surfaces, tidal locking to the host star is likely. Tidal locking makes the planet rotate on its axis once every revolution around the star. As a result one side of the planet would eternally face the star and another side would perpetually face away, creating great extremes of temperature.
For many years, it was believed that life on such planets would be limited to a ring-like region known as the terminator, where the star would always appear on the horizon.[further explanation needed] It was also believed that efficient heat transfer between the sides of the planet necessitates atmospheric circulation of an atmosphere so thick as to disallow photosynthesis. Due to differential heating, it was argued, a tidally locked planet would experience fierce winds with permanent torrential rain at the point directly facing the local star, the subsolar point. In the opinion of one author this makes complex life improbable. Plant life would have to adapt to the constant gale, for example by anchoring securely into the soil and sprouting long flexible leaves that do not snap. Animals would rely on infrared vision, as signaling by calls or scents would be difficult over the din of the planet-wide gale. Underwater life would, however, be protected from fierce winds and flares, and vast blooms of black photosynthetic plankton and algae could support the sea life.
In contrast to the previously bleak picture for life, 1997 studies by Robert Haberle and Manoj Joshi of NASA's Ames Research Center in California have shown that a planet's atmosphere (assuming it included greenhouse gases CO2 and H2O) need only be 100 millibar, or 10% of Earth's atmosphere, for the star's heat to be effectively carried to the night side, a figure well within the bounds of photosynthesis. Research two years later by Martin Heath of Greenwich Community College has shown that seawater, too, could effectively circulate without freezing solid if the ocean basins were deep enough to allow free flow beneath the night side's ice cap. Additionally, a 2010 study concluded that Earth-like water worlds tidally locked to their stars would still have temperatures above 240 K (−33 °C) on the night side. Climate models constructed in 2013 indicate that cloud formation on tidally locked planets would minimize the temperature difference between the day and the night side, greatly improving habitability prospects for red dwarf planets. Further research, including a consideration of the amount of photosynthetically active radiation, has suggested that tidally locked planets in red dwarf systems might at least be habitable for higher plants.
The existence of a permanent day side and night side is not the only potential setback for life around red dwarfs. Tidal heating experienced by planets in the habitable zone of red dwarfs less than 30% of the mass of the Sun may cause them to be "baked out" and become "tidal Venuses."1 Combined with the other impediments to red dwarf habitability,3 this may make the probability of many red dwarfs hosting life as we know it very low compared to other star types.2 There may not even be enough water for habitable planets around many red dwarfs; what little water found on these planets, in particular Earth-sized ones, may be located on the cold night side of the planet. In contrast to the predictions of earlier studies on tidal Venuses, though, this "trapped water" may help to stave off runaway greenhouse effects and improve the habitability of red dwarf systems.
Moons of gas giants within a habitable zone could overcome this problem since they would become tidally locked to their primary and not their star, and thus would experience a day-night cycle. The same principle would apply to double planets, which would likely be tidally locked to each other.
Note however that how quickly tidal locking occurs can depend upon a planet's oceans and even atmosphere, and may mean that tidal locking fails to happen even after many gigayears. Additionally, tidal locking is not the only possible end state of tidal dampening. Mercury, for example, has had sufficient time to tidally lock, but is in a 3:2 spin orbit resonance.
Thus some calculations suggest that a tidally locked would could have temperatures suitable for life on both the day side and the night side.
Another problem with the habitability of a planet tidally locked to a dim star is that many dim stars are flare stars which emit giant flares from time to time. Being in the far side of a tidally locked planet might be safer than being on the near side. But if the flares are strong enough to strip away the entire atmosphere and hydrosphere of the planet, life would die on the far side as well.
So we can assume that your tidally locked planet orbits a dim star which is not a violent flare star.
Part Two: Illumination from companion star or stars.
Possibly the dim star and the tidally locked planet are in a binary or multiple star system with one or more other stars.
Presumably the tidally locked planet would orbit one dim star in what is called a non circumbinary or S-Type orbit And there would be one or more other stars several times as distant.
In non circumbinary planets, if a planet's distance to its primary exceeds about one fifth of the closest approach of the other star, orbital stability is not guaranteed.
So the other star in the system would have to have a closest approach that was at least five times the average distance between the planet and the star it was locked into. And the other star could be tens, or hundreds, or even thousands of times are far from the star the planet orbited.
Depending on the mass, size, and luminosity of the companion star, and on its distance, The companion star might have a visible disc in the sky and appear like a sun, or it might appear as a tiny but brilliant dot of light in the sky.
Depending on the mass, size, and luminosity of the companion star, and on its distance, The companion star might might give the planet a significant percentage of the illumination that the Sun gives to the Earth, or it might give no more illumination to the planet than a star does to the Earth. or might even be too dime to be visible from the planet.
It would be quite easy to design a system where the companion star was as bright as the full moon on Earth, and thus provided enough light for people to perform their activities as well as in daylight. But many times brighter light would be necessary for plants on the day side to be able to grow.
Anyway, the Op says:
The illumination should not come from a star or any single bright source (a moon, ect.)
Which rules out light from a star, or from a planet, or from a moon of a planet.
Part Three: Illumination from the center of a galaxy.
The planet Earth orbits around the center of the galaxy about 26,000 light years from that center, and in the galactic disc of the galaxy. Other things besides stars orbit in the galactic disc, including clouds of gas and dust. And those clouds of dust block out the vast majority of the light from the galactic center - almost all of it.
I once read that if it wasn't for those clouds of dust in the galactic disc the center of the Galaxy would seem several times as bright as the full moon and would be bright enough to read by. That would be bright enough for animals and people to be able to see well, though probably not bright enough for plants to grow. The light would appear to be coming from a glowing region of the sky since the individual stars would be too far away, and individually too dim, to be seen as separate stars, so a diffuse glow would be seen.
The Sun is near the mathematical central plane of the galactic disc. If a star orbited about 500 or 1,000 light years "above" or "below" the central plane, it would be "above" or "below" most of the dust clouds and so would have a much clearer view of the central bulge of the galaxy and the light from tens of billions of stars.
Or maybe the star in your story could orbit the galaxy out in the halo, a spherical region where globular star clusters and isolated stars orbit, and have an even more unobstructed view of the galactic center.
If your fictional planet and star orbited half as far from the galactic center as the Sun and Earth do, the galactic center would be four times as bright as from Earth's distance.
If your fictional planet and star orbited a third as far from the galactic center as the Sun and Earth do, the galactic center would be nine times as bright as from Earth's distance.
If your fictional planet and star orbited a quarter as far from the galactic center as the Sun and Earth do, the galactic center would be sixteen times as bright as from Earth's distance.
If your fictional planet and star orbited a fifth as far from the galactic center as the Sun and Earth do, the galactic center would be twenty five times as bright as from Earth's distance.
But I don't know whether that would be enough light for plants to be able to grown by galaxy light.
The tidally locked planet far side, away from the Star, would be facing more or less toward the galactic center only half of the planet's year. Thus that side would be dark, except for starlight, about half of the planet's year.
The longer than darkness lasted, the more likely the plants would be to die during it.
Fortunately, if a planet orbits close enough to its star to be tidally locked, the planet would be very close to its star and would have a very short year.
Known exoplanets which are believed to be in the habitable zones of their stars and also so close they are probably tidally locked to their stars have years which are tens of Earth days long. Some have years less than 20 Earth days long, and thus if they had a good view of the galactic center or some other light source beyond their star system would alternately face face toward and away from that light sources for less then 10 days at a time.
A few have years less than 10 Earth days long, meaning that they would alternately face toward and away from a light source outside their system for less than 5 Earth days at a time.
The extreme known examples so far are Teegarden b, with a year 4.91 Earth days long, and thus potentially have alternating light and dark periods of 2.445 Earth days long, and TRAPPIST-1 d, having a year 4.05 Earth days long and thus potentially alternating facing toward and away from an external light source for periods of 2.025 Earth days.
Part Four: Illumination by nebula light.
One of the other answers has suggested illumination by the light of a nebula surrounding the star and planet.
Part Five: Illumination by a planet or a brown dwarf.
This would be significantly different from what the OP suggests. Instead of being a tidally locked planet, it would be a tidally locked exomoon of a gas giant exoplanet planet or a brown Dwarf.
A brown dwarf is an object intermediate in mass between a giant planet and a low mass star, massive enough to fuse deuterium but not massive enough to fuse hydrogen. The rough dividing line between massiveplanets and brown dwarfs is about 13 Jupiter masses, while the rough dividing line between massive brown dwarfs and low mass stars should be about 75 to 80 Jupiter masses.
The giant planet or brown dwarf would be a rogue planet or brown dwarf, orbiting the center of the galaxy without any primary star. So your habitable world would be a hypothetical planet sized exomoon if it orbited the rogue giant planet and I don't know what if there is an official term for an object which orbits around a brown dwarf.
I'm sure that the vast majority of exomoons, even planetary sized ones, of rogue planets without stars would be way too cold for life. But some would be heated up by tidal interactions with their planets and with other large exomoons that might be orbiting their planets. It is even considered possible that too much such tidal heating could make an exomoon too hot for life, and thus it seems possible that such tidal heating, when less extreme, could keep a planet sized exomoon warm enough for life, even deep in interstellar space light years from the nearest star.
So an exomoon of a rogue giant planet could be warm enough for life, and have microsopic life forms. But how could it have enough light for plants to grow and to produce an oxygen atmosphere suitable for large animals?. The giant planet and the other moons of the giant planet would reflect starlight, and so there would be a dim light on the surface of the exomoon whenever and wherever one of them was above the horizon, making the surface a little bit brighter than starlight alone.
But that doesn't seem bright enough for plants to grow.
Possibly there will be many thunderstorms in the atmosphere of the gas giant planet. If there are enough thunderstorms at any one time, the combined light of millions and billions and trillions of lighting bolts at the same times might make the giant planet appear to be a huge ball the color of lightening in the sky of the exomoon. And if the sky of the exomoon is hazy enough, possibly the light from the lightening on the giant planet will be scattered all over the sky and appear to be coming from every direction at once, and the giant planet may not be clearly visible.
And that light may be intense enough for plants to grow.
Such an exomoon would be tidally locked to its primary, the giant rogue planet, and so one side would eternally face away from it and never get enough light to grow plants, and the other side would externally face the giant planet and perhaps get enough light for plants to grow. So presumably there would be plant life only on the side facing the giant planet.
And things would be somewhat different if the primary of your world was a brown dwarf instead of a gas giant planet.
The brown dwarf would have a little bit of fusion happening in its core, and would glow with light, but probably almost entirely infra red light and very little if any visible light.
Thus the infrared light from the brown dwarf would help to heat up the orbiting world, along with any possible tidal heating. And possibly both the side that faced the brown dwarf and the side that faced away from the brown dwarf would be warm enough for life.
And possibly the brown dwarf might possibly produce enough visible light from gazillions of thunderstorms for plants to be able to grow on the orbiting world. And possibly the atmospheric haze might scatter the light of the brown dwarf enough to to hide the brown dwarf from visibility. the brighter and more star like the brown dwarf got, the harder it would be for atmospheric haze to hide it.