When I was a kid, I used to do stopmotion - actually got a few ideas to start doing some again. Unless things have changed, you don't really use in-camera (or in-phone) software. You take all your photos individually, and then take them into an editor. If you don't have access to a paid editor, I used to use a program called monkey jam, which renders all your photos into a video, and then take it into a free editor (at the time, windows movie maker) to add sound and other things.
If you're going for a DSLR, I don't think it'd really matter what you got, so I'd just get one of the cheapest ones that are well reviewed. E.g. the 550d/600d. 600d could even be overkill, but if you are at all interested in live action, then it could be useful for that (though only if you are definitely interested, otherwise it's probably wasted money). You're possibly going to want a macro lens, and some form of lighting. If you have a decent dslr, these things are more important, and you could easily spend the same amount on them as the camera itself.
Of course, this adds up pretty fast. On top of that you have the costs of designing and building sets and characters (those materials get expensive pretty quickly).
If you've not tried stop motion before, take Baoliun's suggestion of using your phone at the moment (or if you have an digital camera lying about that your parents brought for their holiday and haven't used since) to try it out. Having a good camera won't help you unless you can create nice looking, but functional sets and characters (it's hard, which is why I never became serious with it), and of course compose a shot nicely (including lighting). Make at least three films with your phone/cheap digital camera and decide if it's for you. I'd avoid dialogue for now if you can.
Personally I love the format but it's hellishly frustrating and takes a lot of patience, so before you invest too much, figure out if it's for you.
links:
http://monkeyjam.org/ (there may be better software for this now, but it's what I used)
http://windows.microsoft.com/en-nz/windows-live/movie-maker#t1=overview (again, there are probably better free editors, but it'll do whatever you need it to do, probably, and is easy enough to use - and doesn't have the learning curve of something like lightworks).