Funny, I discussed the same thing a few months back with a colleague. It could actually use the user's home-directory to drop the file (just like scp does when you don't specify a path).

In fact, the putty project accepts feature requests and comments and tracks those ideas. I would vote for this one definitely ;-)

Here's a link to a similar request, although the project doesn't seem to be that much interested to add the feature :-/

http://www.chiark.greenend.org.uk/~sgtatham/putty/wishlist/drag-drop.html