One other thing to add, the Honda design does not seem to have a fuse for the trim motor. It is directly connected to the battery with a white/yellow wire. The motor will therefore need to have some sort of overload protection built in. Typically, that's handled by a simple bi-metallic thermal switch. That might be at fault too.
You can download an owner's manual here:
http://marine.honda.com/owners/manuals/models/BF40 The back of it has the wiring diagram. That's where I came up with my troubleshooting tips.