Water leaking inside outdoor electrical box. What should I do?

My first impression was that the box isn't really an outdoor model, but on further consideration, it is. You might want to find a replacement at your local home improvement store. If nothing else, try a new cover, as the gasket might be warped, cracked, or even partially missing. Looking at Home Depot online, I see that there is one box that looks similar. That should make it a moderately easily replacement since you wouldn't have to replace the whole light, if that box is the only problem.

You can also look for new conduit connectors for the top and bottom of that box. There may be some gaskets or other seals that have failed that you can't see without tearing the whole thing apart.

Continuing to reread the Question, the existing cover sounds like it has a foam gasket. These can fail without it being a visible failure. Under pressure, they can move due to not being firm anymore, so what looks like a legit seal isn't. I've had them leak even though they look ok. Also, any cracks are a problem. You can try filling it with silicone caulk, but that's a temporary fix, at best. Replacement is definitely recommended. Also, that rusty top screw is a bad sign of a leak. Yes, it's outwards facing, but that rust might not let it seal correctly to the cover.

And as Harper said in in the comment, you might try over-hauling the light itself. It looks simple enough, so it shouldn't be too hard. Disassemble it, replace any O-rings that are bad, replace any plumbers putty, clean out dirt, look for cracks, etc. And this might be a good time for a repaint.

Once you have it disassembled, you should be able to spot anything that looks like it's been recently wet. Anything that's cleaner or dirtier than everywhere else, anything rusty, really just anything out of the norm of the rest of the fixture. If there's glass or plastic, definitely look for cracks or anything broken. Even a lip edge that's cracked might be a place for water to get in.

The part that's coming out of the ground looks to just be standard conduit, so it's not part of the actual light.

Unless the light fixture is actually broken beyond repair, you probably don't have to replace it. With the 12v lights, you could use the same wire that's in the conduit already, but you'll likely have to make some adapters to get it to hook up to the lights. Also, you can probably get an adapter for your timer to send 12v DC to the lights, instead of the current 120v AC. Or you could replace the timer with a newer model with (possibly) better options for the lights.

Many of the newer lights have their own timers, dusk/dawn/light sensors, and are even solar, so you wouldn't need to hook them up to your house. I've even seen them with wireless controls, so you can control them from your phone. As a techie/computer repair/computer programmer, I don't necessarily recommend this. It's not necessarily secure and, really, how often are you going to be messing with your lights that you need yet another app on your phone?

FYI, with there being 2 pieces of conduit coming out of the ground, one is the supply line to this fixture and the other conduit goes to the next light in the yard. You probably have this same set up for all, but the last light. If you end up going with new, solar lights, you could (eventually) rip all of that out. Conduit will eventually fail, causing you problems, even if you don't accidentally nick it with a shovel or, worse, sever it and cause all kinds of problems.

That’s a standard weatherproof (Bell) box, and yes, sometimes they do accumulate and hold water as you have found.

Code not only allows boxes to be drained. Code requires boxes to be arranged to drain; according to 314.15, a 1/8” hole is allowed.

I have found conduit bodies and boxes at a low point (at the bottom of a hill or conveyor) to fill from the bottom, and others to fill from the top. It’s not important, just drill a hole in the bottom and you won’t have any more problems.

So turn off the power and drill a hole and forget about it.

All outdoor boxes should have a drainage hole. These are not left open by the manufacturers because they don't know which way around the box will be mounted. There may be a press-out for the purpose. If not simply drill one or two small holes at the bottom. Make sure they aren't big enough for nesting bees or put some gauze over the hole. This doesn't stop leaks but it stops them causing trouble.