The title pretty much says it all. Sometimes when flipping to my bridge pickup, I get what sounds like a weak, low gain signal. It goes away if I flip the switch back and forth a few times. I'm not sure if it's affecting the other two positions because I don't use them very much (usually keep the neck pickup volume off) and the problem hasn't happened when I've experimented to find out (naturally). I'm guessing it's a bad switch, but I don't know for sure, and I'm just wondering if someone can identify an exact problem based on the symptoms I've described before I start pulling internals out.
Switches, especially Les Paul style toggle switches go bad all the time. Extremely common.

First step, try dusting out with compressed air, or better yet, try spraying some contact cleaner in it. That may very well solve the problem.

If not, you just have to buy a new toggle. Use eBay, they're very cheap.
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Something isn't making contact properly; probably the inner workings of the switch itself (especially if you often leave the switch in the 'bridge' position) though it could possibly be a dodgy solder join around the switch which is being jolted loose/back into place a you move the switch (this is a very rare cause). There's nothing else in a guitar that can cause that problem.

+1 on the contact cleaner and if that doens't work, buy a new switch and wire it up cleanly. It's such a cheap part and they're not built to last so there's no point trying to salvage it if it's really broken.
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I had exactly the same problem with my Washburn WI-64 pickup switch (les paul style).
Just spray it with contact cleaner, it will never make problems again.
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I had this happen to me on my Les Paul, I took it to a tech and for £15 he got it sorted. Common in Les Pauls, not very high quality wiring I assume?