How much atmosphere do you need, really?
A 'friendly' atmosphere with ~20 kPA O2 and overall 90 kPa pressure would be very challenging with Luna's low gravity and her high-insolation orbit.
Useful but less friendly atmospheres are less difficult to engineer.
101 kPa: Earth at sea-level
90 kPa: "Friendly" as per OP
50 kPa: "Unfriendly" - air pressure at the highest permanently inhabited town on Earth
6.3 kPa: "Armstrong Limit" - the lowest pressure at which humans can breath unpressurised
0.6 kPa: "Triple Point of Water" - the lowest pressure at which liquid water can exist in the open
For example, aiming for 10 kPa (1/10th of Earth), humans could walk around the surface in casual clothes plus respirators breathing pure oxygen, giving an oxygen partial pressure comparable to high-altitude areas on Earth (drink plenty of fluids!). Simple plants like mosses and lichens could grow on land, and Earth-normal flora in water bodies.
At 5 kPa, pressurised respirators and elastic counterpressure clothing would be needed (comparable to scuba gear). Swelling (odema) of the hands and face would be an issue on extended stays outside, but could be relieved by returning to (pressurised) habitats.
What gasses to use?
According to https://en.wikipedia.org/wiki/Atmospheric_escape, Earth's atmospheric erosion at present is "charge exchange escape (~60–90%), Jeans escape (~10–40%), and polar wind escape (~10–15%)".
Charge exchange is overwhelmingly just hydrogen and other very light gases. Jean's Escape and Polar Wind Escape are strongly moderated by the gasses' molecular weight and the temperature of the upper atmosphere (exobase). From memory residence time is proportional to the third power of the molecular weight, but I can't locate the reference just now.
The temperature of the upper atmosphere might be the most "easily" engineered variable - incoming solar UV heat's Earth's exobase to around 1000K. If your setting's humans filter most of that out somehow, you could keep Luna's exobase to a moderate temperature, and reduce both Jean's Escape and other methods of loss.
Most of Earth's atmosphere is nitrogen (N2), molecular weight ~32 g/mol, biologically mostly inert. But not useless - it contributes to the overall pressure, and it provides cooling in the event of fire (pure oxygen is a terrifyingly dangerous atmosphere, look up Apollo 1). Your atmosphere needs a buffer gas, and there are many you could choose.
If you have a look at the figure on page 4 of The escape of planetary atmospheres (https://web.archive.org/web/20230220215020/https://geosci.uchicago.edu/~kite/doc/Catling2009.pdf), you will see that Luna could almost hold onto CO2 against Jean's Escape over geological timescales even today. CO2 has a molecular weight of ~44 g/mol, but there are heavier gasses we could use. I will limit my suggestions to those that are non-toxic and non-reactive with common materials, and which provide flame-cooling as good or better than nitrogen.
Xenon is ~131 g/mol, and a noble gas. It cannot be photolysed, it doesn't react with anything except under the most extreme conditions, and it is nontoxic. That would do nicely, but it is rare and exotic, and not found usefully concentrated anywhere in the solar system that we know of, other than the atmospheres of Earth, Venus and Mars.
Sulphur Hexafluoride (SFl6) is manufactured today in thousands of tonnes per year from readily available precursors. Its molecular weight is ~146 g/mol, even heavier than Xenon. It is nontoxic, very resistant to photolysis, and doesn't spontaneously react with anything much except elemental alkali metals. The drawback of SFl6 is that if something does manage to ionise it (x-rays, extreme UV, a high-voltage electrical arc) the decomposition products are toxic fluro-analogs of phosgene. That will react with water if there is any around and form safer compounds, so this might be manageable if high-energy radiation can be limited on your future Luna (orbiting sunshades tuned to filter UV?).
A more fragile but biologically-safer alternative would be octofluoropropane (freon-128). Molecular weight ~188, but breaks down via reactions with UV and water vapor as per https://www.fluorocarbons.org/environment/climate-change/atmospheric-lifetimes/.
Oxygen (O2), CO2 and H2O are the big ones. If you have an open-air biosphere at all, you will have at least 1 kPa of each of O2 and H20, and 0.2 to 1 kPa of CO2. Add another 1 kPa of N2 and/or derivatives such as NO for a sustainable green plant nitrogen cycle. These biologically active gasses will escape over time and need to be replenished due to their low molecular weights (H20; ~18 g/mol, O2: ~32 g/mol, CO2: ~44 g/mol), but the smaller fraction they are of your atmosphere the less problematic this becomes.
My 2 Cents
Yes, a Lunar atmosphere is definitely possible, but consider reducing expectations (and pressure). Orbiting macro-scale sunshades can help by filtering UV.