Actually, since light rail is starting up... South side wouldn't be terrible... but really only if it is close to a light rail location. The station that will be closest to UW is actually by the hospital (not super close to social work building.. but it ain't bad).
So... Along light rail would be... Capital hill, Mount Baker, Columbia City, Rainier Beach... Rainier Beach is the furthest southeast you can get in city. Taking Light rail instead of a bus would be significantly faster (from Rainier to UW stadium would be 30... Cap hill to UW... 4 minutes. That last one is not a typo).
I currently live in the NW part of Seattle and I am fortunate that there is a bus line right outside my house that takes me straight to UW. Its hard to pinpoint all the neighborhoods with direct bus access.. since there are so many... but Wallingford and Ballard are along the 44 route. I am by the current 48 route (Loyal Heights, Crown hill, Greenlake). Pinehurst/Northgate/Roosevelt is another that would be accessible by the 73 or 67 route. I guess what I'm realizing at the end of this long long post... UW is basically "2nd downtown" when it comes to bus service. There is so much stuff that runs through... So there are a lot of opportunities. If you like south side more, start with those neighborhoods first!