Hi M.,
How about a service project? He can probably relate more than most to our service people who are bed bound for weeks. Maybe he could write some commiseration letters to soldiers recovering in the VA.
Another thought, but a little noisy -- a keyboard can be played laying prone, and so can a recorder (transverse flute), especially if he can lay with his head and shoulders over the edge of the bed with the instrument and/or music stand beneath him. This may be a good opportunity for him to discover and develop a musical talent.
Sculpting with clay and pipecleaners are fun, too. Borders and other books, and places like Michaels sell complete craft kits, some of which might work lying down if a table can be placed below him. Starting a model-building hobby might be possible in the prone position, too.
How about giving him a voice recorder and encouraging him to dictate a journal, compose poetry or write some short stories? Some digital recorders can feed directly into a computer for transcription of voice to text. He can even publish if his stuff is good enough.
Encourage him to make up his own games. If he comes up with good ones, use it as a starting point for putting together a book of games for the bed bound or a boxed game for the bed bound. It's never too young to become an entrepreneur!
The BSA badge idea is a good one, too. At 11 I'm guessing that he has just left Webelos for Scouts. Much of the material for Tenderfoot involves studying from the manual. Now's a good time! He can also use the time to plan out what activities he will want to undertake for badge work once he's up and about again. If he's still a Webelo, he can work on things like the religious awards and academic based activity pins (sounds like sports will be out for a while).
I wish your godson a fully and speedy recovery!
R.