You could use a 12VDC timer, but finding an electric timer that runs off of 12V is hard to come by (24VDC is common for industrial/automation).
Another idea is to use a larger capacitor. This may not produce the result you want, the light would start out bright, but dim to nothing over a certain amount of time, depending on the size of the capacitor.
Go to Radio Shack, and ask those guys if they have anything in all their little drawers for a timer or timing relay (Time-on delay) that could be powered off of 12VDC.
Good luck... I looked for a timer online awhile back, and didn't find anything, but didn't put too much effort into it either
Oh yeah, I have seen many seatbelt lights that stay on for a short period of time after the vehicle has been started, and some that stay illuminated till the seat belt is buckled.
EE specializing in Automated Control Systems