I don't call myself a hostel person, but there were times I used hostels a lot back in my early 20ies, actually when I first arrived to Tbilisi back in 2011 I stayed exactly at hostel. Hostels are good for couple of reasons - you can meet new friends, but what's important they are cheap. Tbilisi are no expectation when speaking of cheap stay, there are dozens (if not hundreds) of budget price hostels.
The cheapest ones comes as low as 7.5 GEL per person per night typically it means you will get a bed in a dorm room and in most cases will be located out of city center.
There have been two times I have stayed at Tbilisi hostels, and in both cases I was looking for a separate room, first time I got it for 40 GEL in Avlabari district, the second time I was using Tbilisi hostel located on Rustaveli avenue and it cost me 80 lari for a separate room. If you prefer single room - you should expect to pay at least 30 Lari for stay in Tbilisi hostel.
10 GEL = 4.18 USD / 3.84 EUR
In this list you will find 71 Tbilisi hostels listed from lowest to highest price available