My agent picked me up when my book wasn't just unfinished, it was unstarted. She really liked my art, and the weird little desciptions that went with it, she'd read part of my webcomic, and she came to a particular painting where I'd mentioned a children's story idea and said "Did you ever write this?"
"No..." I said, and then in a fit of mad bravado, "but I could!"
"Do it!" she said.
About two months later I gave her a manuscript, and about three months after that, she sold it. This could have failed at any stage--I could have flaked, she could have gotten something that she wasn't equipped to or interested in selling, no one could have wanted it, I could have turned out to be a psycho who called her non-stop--but arguably, that's pretty much the gamble agents take with practically anything, so it was really only one more layer of risk than the usual.
My experience is madly atypical, obviously, and I don't suggest reading too much into it, but from a sample size of one, my guess is that sometimes agents will play a hunch. I suspect some of those unfinished manuscripts fall in that category--the agent reads it, goes "Hot damn! Yes!" and snags it, because they're either excited enough or have seen enough evidence to gamble that the author can pull it off in the end.
It's like horse racing. Most of the time you bet based on the horse's past record or the track condition or the bloodlines or hot tips or whatever, but every now and again you drop some cash on the one with the same name as your first-grade teacher, just in case it's a sign.