I doubt that anything like this would ever be done, but I find it theoretically interesting.
Much has been said about the undemocratic nature of Senate appointments. Well, we live in a representative democracy, so Governor Paterson has in close proximity some representatives of the people in the form of the state legislature.
I propose that the New York Democratic state legislators be gathered in one place. The governor gives them a shortlist of five names. The legislators then vote, with run-offs until a candidate has a majority of votes. This can either be done with separate rounds of voting (with caucusing in-between as people try to gain votes?) or by instant run-off voting (IRV). Maybe the candidates get some speaking time before votes.
This method has several advantages. It is theoretically more democratic. Sure, Republican districts get the shaft but, yeah, whatever. It demonstrates the potential effectiveness of requiring run-offs (and IRV, if used). It increases transparency, assuming results are released, as we know who was considered. If the shortlist is released in advance with enough time, the politicians field input from their constituents. Paterson can keep his hands clean and not find himself in opposition to an embittered faction unhappy that their candidate wasn't selected.
In effect, this would be a Democratic caucus. Ideally, outside observers, including media, would be invited so that it is not some huge backroom deal. The process has flaws, of course, but I think it sounds better than having the governor appoint or a lame duck caretaker. It also has the potential for being an enlightening and entertaining spectacle. I'm sure they would never do it, but it's something that's a bit outside the box.