Have you looked at the Friendship/Bloomfield/Lawrenceville area? They're more young and "hipster" areas and absolutely accessible by bus. If you're looking for something cheap, stay away from Shadyside and Squirrel Hill. They are definitely trendier areas and you will pay for it. You could potentially find places on the South Side that are within range, but I think your best bet is to stay in the East End.
I just looked quickly at Craigslist and found these places:
http://pittsburgh.craigslist.org/apa/4383796126.html
http://pittsburgh.craigslist.org/apa/4355106098.html
The University has some apartments on Oakland Ave and Atwood St, which would be maybe a 15-20 minute walk to BST. I lived in one on Oakland Ave my senior year of college, and, though it wasn't terribly modern, it was very clean and well-maintained. If you don't mind living in South Oakland, which is sort of dirty (generally) and innundated with undergrads, they would be a good choice. Again, Oakland=dirty, university apartments=not dirty. I think there is value in living within walking distance of the lab.
http://www.ocl.pitt.edu/university-owned