Marie,
Monterosso would be a great choice as it's the largest town, has the greatest number of accommodations and restaurants, and the best beaches of the five towns.
You might find this list helpful.....
http://rebuildmonterosso.com/visitors-information/
I'm not sure how current it is, but you can try contacting the businesses directly. A couple of places you could look at.....
I haven't stayed at either of these but they both appear to have rooms which may be suitable for families. Check Trip Advisor for reviews.
Hotel Porto Roca also has larger rooms, although this tends to be more at the top end of the price scale - http://www.portoroca.it/en/index.php . They also have a beautiful swimming pool overlooking the ocean as well as great views from the hotel. It's also at the far end of town, so you wouldn't be bothered by the cruise ship hordes as much.
Whatever hotel and town you decide on, it would be prudent to get a booking soon as July is peak travel season!