When I was in Stuttgart, I stayed at Alex 30, which was named Alex 30 cos the address is Alexanderstrasse 30. Very easy to get to from the hbf, just take any line of the u-bahn that stops at Olgaeck (#5 and #6 and one other line I believe), and you're right there.
Beds were SO COMFY! It's also restaurant and a bar and this little shop too. Owners are friendly and knowledgeable.
Wasn't really a big party hostel though, didn't strike me as one anyway. I was there on a Monday though, but the bar was partially crowded but it just seemed kinda chill more so than wild and crazy.
You can rent bikes there too, and they have internet and you have to pay for it but it isn't too outrageous.
Link to their website: http://www.alex30-hostel.com/