Where to stay in Venice?

I am not sure what you mean by "small island". Venice is a very walkable city and i would book a hotel based on your budget and try to find a good location for you. I like Dorsoduro neighborhood because it's a bit off the beaten path and has some pretty nice piazzas. Check Booking.com and vrbo.com for additional hotel or apartment options. For things to do in Venice, here are just a few:

Take a vaporetto the length of the Grand Canal (from the train station to St. Mark's square)
Tour the Doge's Palace
See St. Mark's basilica and the Frari Church
Accademia and Guggenheim Museums
Vaporetto to Murano for a glass blowing demonstration
Vaporetto to Burano for lace making
Wander the back lanes, find a nice outdoor cafe and enjoy some vino

Enjoy!

If what you mean by "center of the city" is the city of Mestre, which is one train stop away on the mainland, the answer depends on what YOU decide you want to see in Venice. Check out some guidebooks, then decide.

I second Stephen's good advice. The Dorsoduro has the feeling of "off the beaten path" but is really quite close to everything.

Most people do not want to waste valuable time with a commute from Mestre (mainland) or Lido or Murano or whatever island you may be referring to.

Booking.com does use the term "Venice City Center" to refer to all the areas closest to sightseeing and they don't have anything under $200 US per night using that designation. If you just use their search for "Venice," you get a few cheaper options in Santa Croce or Cannaregio areas; check their map locations. I'm guessing (only) that prices could come down closer to your stay if rooms are still not sold or somebody has a cancellation. For instance, if you search for a similar period next week, they have a few cheaper listings that were marked down.

We stayed here in Venice and loved it. Less than $100 a night, and a 2 minute walk to Rialto Bridge near EVERYTHING.

http://www.cabragadin.com/

I agree with Stephen and Chani. Staying in Mestre is not worthy whereas Dorsoduro district is quiet and close to the center. Furthermore if you are travelling alone you might want to consider staying in a b&b or in a rental flat if you are going with your family. We stayed here http://veniceapartmentrentals.com/ in an apartment with a view on a nice canal.

We recently stayed at Locanda La Corte and liked it very much. Very good location. I would say it is worth paying extra, though, to be in Venice "City Center."

I recommend Rick's suggested Locanda Casa Querini. Just left there yesterday! Close to the action, but on a quiet piazza.

Check out Hotel AL Codega, great location above average accommodations reasonably priced.
Lorieann