I think, to answer the question in the OP, that the reason they signed Tebow was this: Most teams have a 3rd string QB. The Pats, with absolutely no risk to themselves in any way, thought that perhaps he might be a better 3rd string QB than what they had, because:
(1) Tebow brings something a little different than a traditional QB,
(2) like it or not, Tebow *has* won in the NFL, and winning experience helps,
(3) Tebow can also do other things for you (potentially) so you get a little more than just a 3rd string QB, and
(4) if they have to go with their 3rd string QB for a long time they're screwed anyway; maybe Tebow could hold down the fort for a short time period better than their existing QB3.
I really think it's that simple.