If you have a kid ready for a significant hike, the Loop Trail on Tumbledown is a great one to consider. The loop trail is moderate for adult hikers and challenging for kids. There's actually a sign on top that says the hike is not recommended for kids and dogs, so this would only be one to attempt after gaining confidence on smaller hikes. My kids were around 7 the first time I took them up.
Although it's a hard hike, the reason I recommend it is there's lots of small rewards along the way. The trail starts in the woods before changing to rock scrambling. Some larger rocks will present a fun challenge for kids to figure out. Then there's the chimney and the short vertical rock tunnel (charmingly named "Fat Man's Misery") you have to climb using metal rungs. Very exhilarating. Once you're above tree line it's an easy walk along a long ridge. If you go in mid to late August you'll find wild blueberries all over the ridge.
I'll confess we never take the offshoot trail to the true summit, because by the time we get to the junction, our feet are begging for the cool waters of Tumbledown Pond. We head straight to the gorgeous mountain-top pond where the kids swim (brrrrrrr) and enjoy whatever sweet treat we bribed them with. We do this hike every year while camping at the fabulous Mount Blue State Park. Even if you don't hike Tumbledown, Mt Blue State Park is THE place to go with kids.