Do you know a good and inexpensive hotel near the city center of Boston?

no

I recommend trying Hostelling International USA.
I did this in NYC and it was wonderful; it's not just for the young, and not just for international tourists!
I notice current Boston price is only $32, and they have a number of amenities!
Here is a link for 2 hostels in Boston:
http://www.hiusa.org/usa-hostels/ma/boston.shtml

I was in Boston in February, and am going again in August. I am from Boston, but stationed overseas with the military, so I visit as often as I can. Anyway, I always use www.hotwire.com when I'm trying to book a hotel. The last time I visited, I stayed at the ExtendedStay Boston, in Danvers, which is about a 10-15 minute drive from the heart of Boston. It cost $42 per night. When I visit in August, I'll be staying at the Sheraton Ferncroft, also in Danvers, and paid $43 per night. Hope this helps.

Have you used Tripadvisor to do a little research? I'm guessing that by the city center you mean Back Bay/Downtown area, correct?

The 463 Beacon Street Guest House has pretty low rates as well as the Commonwealth Courtyard. THey are a steap above a HOSTEL but will do the job at under 100 dollars a night http://www.tripadvisor.com/Hotel_Review-g60745-d123197-Reviews-Commonwealth_Court_Guest_House-Boston_Massachusetts.html
http://www.tripadvisor.com/Hotel_Review-g60745-d120999-Reviews-463_Beacon_Street_Guest_House-Boston_Massachusetts.html

The Charlesmark Hotel also has good reviews and it's located in a great location, too. http://www.tripadvisor.com/Hotel_Review-g60745-d225873-Reviews-Charlesmark_Hotel-Boston_Massachusetts.html Around 100$ a night.

The Hi Hostel on Hemmingway Street is located in the Back Bay--best hostel in Boston and the best location. Dorm like accomodations--super cheap--http://www.tripadvisor.com/Hotel_Review-g60745-d114141-Reviews-Hostelling_International_Boston-Boston_Massachusetts.html

I agree with NO

www.vrbo.com