Envoy has a great location just above Meidan and is one of Tbilisi's best hostels. It's an old parquet-floored house, with six dorms, four private rooms, a wonderful roof terrace with superb Old Town views, a good modern kitchen-dining area, and an adequate number of showers and toilets. Do try to avoid the one double room with no windows, though.
Guests get a free two-hour Old Town walking tour, and on Saturdays it runs a minibus to its Yerevan hostel with stops at several of northern Armenia's main sights (140 GEL for guests, 11 hours).