There are no real hotels in the vicinity of Sandy Hook. You would have to travel south to Long Branch, east to Eatontown, or north to Middletown for any of the big chain hotels.
My sister tries to go there each weekend with her kids. They generally leave the house around 8AM to make sure they get into the park before it closes. Like some of the PP's stated, it gets VERY,VERY CROWDED! Combine that with a major bridge replacement project currently going on just outside the park entrance, and you can have a big mess on your hands. I just drove past the entrance yesterday on my way down to see some dolphins that are currently swimming in the Shrewsbury River near the park. I was shocked at the work that was going on.
I would recommened Pt. Pleasant Beach as a nice location. Some small local hotels there, plus B&B's. Don't know the policies on kids at the B&B's however. You will probably run into 2 night minimum requirements at any hotel you look into down the Jersey Shore at this time of year (3 on holiday weekends) . Nice boardwalk, amusement areas, and a small aquarium in Pt. Beach.
The Lavallette area is also very nice. Much quieter then Pt. Beach and Seaside. Very nice family beaches, there is a boardwalk, but there are no rides on it. It is truly boards that you walk on. I used to live in Lavallette, so I am a bit partial to it.
Seaside Heights can be a bit wild on the weekend nights. If you like people watching, there are planty of characters to see down there! The boardwalk does have a lot of things for the kids to do, and there is a water park as well. Personally, I would try to stay in Seaside Park area (much quieter), and go to the Heights for the Boardwalk.
Here is one last idea for a day trip, which may be even closer to home. Keansburg Amusement Park. They have an awesome water park, and a really nice amusement area for kids. The guy who currently owns the place bought it back after it was neglected for years by other owners. His grandfather was one of the original owners many, many moons ago. You can do the water park for a few hours, then head across the street to the amusement park. It is on the bay beach, but on a clear day there are great views of NYC !