This is a very basic list without much info but it should get you started on places you could stay! 
I must also mention that since the University is very centralised in Galway, the max you would have to walk from the majority of these places is 20mins.
To use a reference, it takes 15-20minutes to get from the bus station in Galway to the university.
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Sleepzone
www.sleepzone.ie
091 566999
Bothar na mBan
Galway, Ireland
Snoozles
www.snoozleshostelgalway.ie
091 530064
College Rd.
Galway
Nimmos Hostel
http://nimmoshostel.com/
091 586661
1 Upper Dominick Street
Kinlay House Hostel
www.kinlaygalway.ie
091 565244
Merchants Road
Eyre Square
Galway City Hostel
www.galwaycityhostel.com
091 566959
Frenchville Lane
Eyre Square
Claddagh Hostel
091 533555
Queen Street
Eyre Square
Barnacles Quay Street House
www.barnacles.ie
091 568 644
Quay Street
