For example, a quote from one of the Witcher 3: Blood and Wine's side quests says
The eggs themselves are a sight to behold, luminescent.
The man (hologram) is referring to giant centipede eggs.
For any other world, is there a good reason why eggs would glow? For centipedes?
The species is parasitic in at least one stage of life. Glowing eggs attract creatures that want to eat the egg (since eggs have lots of fat in them, and don't run away or fight). (Thanks to an excellent comment by @HenryTaylor.)
Thus, the "egg" isn't really an egg like we think of them, but actually bait for transfer to a host for the next stage of life.
After it grows a bit, it leaves the host to grow to final adulthood and then start the cycle over again.
On Earth poisonous animals are often brightly coloured, e.g. Poison dart frogs. Perhaps bioluminescence evolved as a signal that something is poisonous, and the eggs are poisonous to predators, or pretending to be.
Let's assume bio-luminescence is rare and not the norm: if the environment is full of bioluminescent rocks, then it would be more a camouflage. So back to basics: bio-luminescence makes things visible, and usually more recognizable and identifiable. How could this provide any advantage to an egg ?
Identification for better protection. If from a social specie, or at least one taking great care of its eggs, the eggs would be better nurtured. In this case, the bio-luminescence could also be used as a way to communicate the egg's needs (warmth, moisture, etc.).
Identification as a warning. The eggs could be highly toxic and poisonous for predators, or have other defense mechanisms (an electric egg ?). This special signature would keep predators at distance, like some animals using very visible colors.
Identification as a trap. The eggs are easily spotted by some wannabe-predators, only to see the latter being stick to the egg and dissolve by some digestive liquid dripping from the egg. Or the egg need to be eaten to reach a parasitic form that is an intermediate step in its growth). This could also be combined with 1. where care-taking adults use the eggs to attract preys closer.
Perhaps the egg laying creature is the apex predator in its world, with such a overwhelming predatory advantage over all other life, that even its' newborn offspring can defeat anything which might be attracted by the light. The glowing eggs might serve to provide its occupant with a first meal.
This article https://www.allaboutbirds.org/the-beauty-and-biology-of-egg-color/ provides a number of interesting examples from bird eggs which suggests reasons why conspicuous eggs might not always be a disadvantage. I'll summarise some of the most relevant.
1. Group nesting
Several Great Tinamous lay their bright blue shinny eggs in a single scrape in the ground. The eggs are not camouflaged despite the risk of predation. It is suggested the bright colour of a Tinamou egg attracts other Tinamous to lay in the same nest. It is theorised that the larger number of eggs means it is less likely a predator will be able to consume all the eggs in the nest.
This could be a good fit for your giant centipedes, especially if they are a species which already produces a large number of eggs, with a small proportion expected to survive.
2. Parasite protection
A number of species have highly patterned eggs to distinguish them from brood parasites like cuckoos.
Bioluminescence could be a signature which is hard for the parasite species to mimic. However this would require that the parent centipedes give sufficient brood care to attract such parasitism. This benefit would have to be weighed against the cost of the increased visibility to predators.
3. Improved visibility
Hole nesting birds often have quite conspicuous white eggs. This may simply be because there is no need for pigmentation for camouflage as the eggs are already concealed in a hole, however it may also help the parents to see the eggs in the dark.
The centipedes may nest in caverns or other enclosed nest holes, where predators spotting the eggs is not an issue. The bioluminescence may help the centipedes to see their eggs, and care for them, or simply avoid accidentally damaging them.
Additional Notes
Quantula striata, A terestrial snail species, actually does produce bioluminescent eggs.
An additional suggestion (which I have no real world example to back up) is that the luminescence could simply be to protect the eggs from accidental damage. Perhaps the greatest risk is that the eggs might be trampled by some careless herbivore. Any such creatures might be very willing to avoid any sign of the giant centipedes.
If the parents give nest-care the luminescence (the color and/or the intensity) could be tied to the temperature of the eggs.
So the parents can tell whether the eggs are too warm or too cold and can take action accordingly.
Maybe they don't glow all the time. Several insect proteins glow under black light, it's just a side effect of the protein structure. And the eggshells of insects are made of proteins.
Alternatively it could be a similar side effect and they glow all the time, but you could make them a burrowing species. Since the eggs are buried fairly deep it doesn't matter if they glow, or at least it does not matter enough.
The luminescence tells other giant centipedes there's a large hungry female nearby and she's going to be very defensive of her nest, stay away.
The luminescence tells the mother not to eat her own eggs no matter how hungry she gets, maybe it triggers an instinctual suppression of hunger.
Other animals may be attracted to the luminescence which would normally be disadvantageous for the eggs except there's a ravenous giant centipede waiting to ambush anything that gets too close.
Giant centipedes clearly use magic in their metabolism — otherwise they’re too big for a centipede’s rather primitive respiration to get enough oxygen to their internal tissues to move that fast. And since we know nothing about how that magic actually works, perhaps luminous eggs are an inevitable side effect of it.
The largest threat is creatures that lay their eggs in another species nests like the coo-coo. Glowing eggs are a signature that other species can't mimic.
The eggs absorb their nutrients from flies that land on the egg and are digested through the membrane.
Other creatures will protect the egg as it provides a light source to allow them to hunt in the dark.
The light will help a small ecosystem grow in the nest for the young centipedes.
Well, let's have a look at what bioluminescence is for in the natural world.
It can be used to communicate, but eggs are inanimate, so that's probably not an option. It can be used to ward off predators, but really - some glowy lights on a non-moving, nutritious rock isn't going to scare away predators. But, however, bioluminescence can be used to attract things, so the only reason for it would be if the eggs wanted to get eaten. By this, as another respondent has suggested as well, I mean that your creature's offspring are parasitic. But they can't harm the predators, or else they would learn to avoid the creature's eggs. It could be a kind of symbiosis, with the predator getting nutrition, and the parasite getting something else in return.
Bioluminescence is actually used as a form of camouflage by way of counter-illumination (https://en.wikipedia.org/wiki/Counter-illumination).
Here on Earth (I think) this is exclusively used by marine animals to avoid detection from below, i.e., the bioluminescence mimics the surface of the water so, when viewed from beneath, they avoid displaying a conspicuous dark spot to predators.
Perhaps the eggs are normally laid in areas with numerous bioluminescent plants and/or fungi and the glow prevents them from displaying the aforementioned conspicuous dark spot?
Some species of fish lay eggs across a very large area. It could be possible that a creature with similar behavior would use glowing of eggs to locate it's own eggs.
Perhaps even the species lives in a group in which babies are cared for by a dedicated sub-group (similar to bees). In this case the ability to notice and locate eggs would be even more important as the those taking care of the babies haven't laid them - and thus won't have knowledge of their location.
That said, I still think in the advantage glowing eggs gives a predator is more significant, and would make the property mainly a liability. We can also see that almost no (to my knowledge no) species on earth lays such eggs.
Perhaps the eggs are deposited in an environment where the only true threat comes from fly larvae hatching and eating away at it? (Say it's poisonous for vertebrates, unreachable for most invertebrates) There's research that certain wavelengths of visible light will significantly lengthen or even prevent hatching in fruit flies.(https://www.ncbi.nlm.nih.gov/pubmed/28578425)
It has been theorized that the main reason for Zebras colouration is to avoid certain files rather than other more prominent predatory animals.
Could the luminescence be protection from some (as yet) unknown predatory animal. It would have to have sensitivity to light, making it nocturnal. A light source in the dark would be a good defence against it.
Here's a theory: it could glow at the very end, right before it hatches. Now, if nothing else really glows, then it is super attractive. Like a homing beacon for a nutritious breakfast. Some animal comes up to eat what looks like a easy meal. But suddenly, the egg cracks open and the animal inside the egg is more dangerous then the one that came to eat it. The newly hatched animal gets it first meal.
Perhaps the females lay unfertilized eggs, which then begin to glow. Males are attracted to the glow and fertilize them. Usually in fertilized eggs the glow starts to diminish but it can last for quite a while after the male has visited.
I am reminded of the kakapo parrot. Especially of the description of its mating ritual by Douglas Adams in his book "Last Chance to see".
Adams describes hilariously, but also very convincingly, the kakapos mating ritual as "incredibly complex, very long drawn out and almost completely ineffective".
Adams goes on to elaborate several different aspects of the mating ritual, all of which seem indeed more suited to prevent mating than supporting it.
Strange, you might think, but it is actually a survival trait of the kakapo.
Because the biggest threat to the survival of the kakapo species as a whole is not being eaten, but eating all available food. Mating and having to many offsprings, in the face of an almost complete lack of predators, would lead to a possible collapse of the kakapos niche in the ecosystem.
So the kakapo has to dampen the growth of its own number in order to prevent starvation and the anti-mating ritual is the mechanism to achieve that.
Please refer to the book by Adams, because of his superior writing and for any details.
Translating this to the luminescent eggs, I could imagine a species which eats the eggs from the nests of other breeding parent animals; not its own eggs of course, prevented by e.g. pheromones.
The result would be an efficient mechanism to ensure that eggs in a nest can only survive outside of eyesight of any other member of the species. This in turn would ensure a maximum of density of breeding parents in any given area.
This assumes of course that the disadvantage of glowing in the dark is somehow evened out, e.g. by any of the mechanisms described in other answers to this question.
In short, being very visibly edible could be the survival mechanism in an environment which is otherwise in danger of collapsing due to overeating species.
To add to Skye's answer:
This is what they look like with their webbings:
So we have a similar case in our own real world. Sometimes a juvenile being glows in order to attract food. A glowing egg may be the ticket to a baby's first meal.
Do note that glowing need not be an advantage, it need merely not be a significant disadvantage. Suppose the adults are luminescent as well and use the light to hunt in the dark. That could be a big advantage. If the glow gets turned on prior to hatching that would make the eggs light up, but that would have to provide a larger disadvantage to the eggs' survival than the advantage to the adults' hunting abilities to prevent the trait from taking off. Barring one of the additional advantages to the glow mentioned by other posters, the glow of the egg is likely to disappear over the next few million years due to the additional incremental advantage of not glowing as an egg, but it hasn't happened yet, so you still have glowy eggs around.
I would think that hungry, venomous, giant centipede parents hanging around the nest and eating anything that comes close could easily be enough of a deterrent to predators to render the visibility of the eggs insignificant to the survival of the species.
I am surprised that the obvious answer is conspicuously missing. Perhaps it is camouflaged by being so obvious.
The entire environment is bioluminescent.
If the eggs were not, they would stand out for their lack of bioluminescence. That is, they would be an obvious dark spot in a bright background.
Camouflage only works if it is the same as the background.
Think of the world of Avatar.
Maybe the glow is a byproduct of a chemical that the larvae need. The young need handwavium upon birth, and having handwavium in the eggshell provides them a good first meal. The only problem is that handwavium happens to glow.
Maybe the eggs are laid near a species of toxic, glowing mushrooms. Predators confuse the eggs for mushrooms and don't eat them. Herbivores think that they're just as toxic as the mushrooms they look like, so they don't eat them.
Maybe the bugs have a weird symbiosis with humans/the dominant species. The glow of the eggs attracts humans who take care of them, and when the eggs hatch, they serve as good pest killers or they eat the dust in people's homes.
Maybe they live in a world where their primary predators are blind, and the glow lets the parents see them to protect them without giving away a scent to attract the predators.
