Westport: We stayed at the Wyatt Hotel. It had location, location, location. What we didn't get was a good night's sleep. Our rooms were up over the bar/ restaurant area and it was loud until late. At 2 am, one of the hotel's workers came and parked himself under our window and began a loud cell phone call. The breakfast was also very, very sad. The cost/value ratio was extremely poor. This is not a hotel I would ever stay at again.
Belfast: Not a "charming" hotel, but the location can't be beat if you are going to be exploring the Titanic Exhibit. Premier Inn Titanic Quarter. Less than 5 minutes to the Titanic Exhibit walking along the river and about 15 minutes to downtown walking, less by bus. The rates here are a good value.
Instead of Portrush, consider Port Ballintrae just a little further along the coast and the Bayview Hotel. This was our absolute favorite hotel of our trip. They also have two apartments that they rent out that might fit a family trip quite well. The view out over the harbor is magnificient. This is a town where Irish people come on holiday and there are far fewer American tourists. Public Transport is good with the bus stopping right outside the hotel. It is a 10 minute walk to Bushmills. We walked to the Giants Causeway on a lovely path over the near by beach and headlands. It would be a great family walk. Dunluce Castle and Portrush were also quite walkable. I do confess we took the bus back from Portrush!
Here is the hotel's website
http://www.bayviewhotelni.com/