A problem I've run into with taillight bulbs is corrosion in the socket itself. Generally they're pretty well sealed against rain and water, but with age those seals can fail and allow moisture in.
As Rodney stated in his answer, the first thing to try is a new bulb; they can look fine and even power up momentarily but go out when they heat up, so at a minimum you should at least swap the bulb from the other side to verify the bulb. Use a piece of sandpaper to clean the contact on the bottom of the bulb to make sure there isn't any corrosion on it.
Assuming it still doesn't work, remove the bulb and inspect the inside of the socket. There will be a spring contact at the bottom and a metal case that holds the bulb itself; look for any corrosion, a white powdery-appearing substance, on both parts. If you find any, use sandpaper to clean it thoroughly, then check the spring contact to make sure it's not bent down to where it doesn't make a good connection when the bulb is inserted. If you compare it with another socket, they should both stick up the same amount. If not, bend it up manually to match.
That usually fixes the problem for me, so hopefully it will take care of yours.
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