I've never been to L'viv and I'm a little overwhelmed by all the hotel options. I'd like to stay in a hotel that is central and gives me good sense of what L'viv is like. I'd love to know where locals tell their friends to stay when they visit, and why those areas and hotels are special. thanks!
But yeah, Darya is right - you can easily stay in the historic centre on the cheap, so try and aim for that. L'viv is a great city, and the best way to see it is on foot, while staying as close to its heart as you can.
My friends stay in hostels right on the Market Square Apartment or somewhere close to it. This is a cheap and comfortable option, especially if you are not traveling alone. You can choose a hostel in 1-2 bus stops, too, because historical center is rather big.