Well, if you want to investigate, this is a map I've compiled over time with the BBs and hostals (hotels with basic ammenities, different from "hostels") most recommended by visitors at the Barcelona TripAdvisor forum.
http://bit.ly/hostals_and_BBs_in_BCN
And this would be a map of districts of the city: http://bit.ly/BCNdistricts (and their hoods: http://bit.ly/BCNhoods) so you can get your bearings. Albeit Barcelona has different 'centres' so to speak, the one considered "the centre" would be Plaça Catalunya, in the border of l'Eixample and Ciutat Vella. These two districts do concentrate about 75% of accomodations chosen by visitors. Note though that the city grew out of absorbing exisiting neighbouring towns, each one with its own centre, shops, etc. so outside the above mentioned districts you have plenty of choices that are not residential-only areas but areas with life of its own.
Btw, albeit Barcelona receives tourism all year long, mid May is not a particularly "troublesome" time for visiting the city, you should find good accommodation in the lower range of your budget.
As per boutique hotels -these are never too cheap!- you could have a look at a few here:
http://www.greatsmallhotels.com/spain/barcelona/boutique-hotels
..and if you're looking to splurge yourself, check these: http://intelligenttravel.nationalgeographic.com/2013/01/24/the-10-best-hotels-in-barcelona-afitz/ but note these are not within the budget you mentioned.