Here are some of the hostels I checked out (not staying) while walking around town.

Bašačaršilja M.M. Bašekije 65, Sarajevo
+387 33 535 829 (phone)
+387 33 232 109 (fax)
061 131 813 (mobile)

I saw the price as low as 6 euros/person/bed/night in a 8-bed dorm on the price list sticking up the window. However, you might not see this price on the website. It is in Bosnian, but you might be able to figure it out. Any way, email or give them a call. Remember to ask for the 6 euros room.

M.M. Bašekije 63, Sarajevo
+387 33 238 680 (fax)
061 800 263 (mobile)

As low as 10 euros/person/bed/night

ZemZem Caffee -Pizzeria i Pansion
Mula Mustafe Bašeskje 61
033 239 648
061 804 200
20 euros per person in private room. 50 euros for an apartment up to 6 people.

For more

The link provides a list of hostels and some hotels in Sarajevo. You can find a few for really cheap, for example, as low as 5 euros at Hostel Marin Dvor. However, I smell something fishy about this place. How can it be that cheap given that the living cost in Sarajevo is comparatively high.