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NH has some very specific rules about youths operating OHRVs. The important information can be found on page 10 (sheet 12 of the pdf) of this document: http://www.eregulations.com/wp-conte...13NHATV_LR.pdf
I believe you said your son is 11 years old which means he isn't required to take an OHRV safety class. He will be required to do so when he turns 12 but they are offered for free. As an 11 year old, he obviously has to be under your supervision but he is not allowed to cross public roads. Once he is 12, he still needs to be supervised by you but he'll be able to cross roads and ride along approved ones legally.
The closest trail to you is the Rockingham Rail Trail. It's 12ish miles long and has lots of road crossings which makes it unsuitable for an 11 year old. The trail is mostly straight and there are very few if any challenges so it's easy to get bored on it.
If you venture to the Dunbarton NH area, there is the Hopkinton Everett Riding Area (aka Clough state park). It has plenty of double track trails to explore and a couple more challenging single track trails. Technically there are a few road crossings here as well but they are woods roads as opposed to high traffic paved roads. Lots of NESR people ride here throughout the season.
For whatever reason, the Little Monadnock Family Trails area is not listed on the 2014 NH OHRV Map but it is open to anyone with a NH OHRV sticker. They are farther West in Richmond, NH. I've only quickly explored the area but there are a lot of trails of varying difficulty and you could easily spend a few days exploring without repeating trails. https://www.lmftnh.com/home.html
Pisgah State Park is in the Southwest corner of NH and has trails for OHRV's. I haven't had the chance to explore that area yet.