It could be that a PO used the wire from your license lamp to power the brown wire (running/parking lights), leaving you with no wire to hook the license lamp to.
An easy solution that would still leave your standard 4-wire trailer hookup functioning would be to buy the other half (any parts store, even Wal-Mart), then tape/heat shrink all the others but the brown one on your new piece, then run the brown wire to your license lamp.
Then you could hook to most non-brake-equipped trailers with your connector by unplugging your license light and plugging in the trailer. Having the connectors mated up all the time will keep the connections cleaner, too.
SSOW
