some places in NE Ohio
To add to Gary's suggestions- I grew up real close to Mosquito SP and it's OK. Quiet place, ranger makes sure of that. Many tree filled sites, Ice cream, shopping centers, etc. close by. Lake is man made, so the bottom is silt. Jet ski rental close to park. Haven't been to Pymatuming in 20 years or so, but remember having a good time there.
I'm not sure if their are any campgrounds in the area or what they may be like, but Geneva on the lake can be a pretty fun place as well. There's some small amusement parks, wineries, and restaurants along the lake. Headlands beach is huge and fun if you like swimming. It's around the NE corner of Ohio in Ashtabula county. I'd do a little research on the area and contact some of the campgrounds to see how they run their business. (Family sytle, more for bikers, etc) The area has been known to be rowdy at certian times of the year, but there are things more family oriented. Maybe someone who has been there recently will chime in with a good campground to set up at. Have fun!! --dave