Your home state of Rhode Island and the state of New Hampshire are both members of the Drivers License Compact (DLC) so if you are convicted of the NH speeding ticket the courts there should inform the RI Division of Motor Vehicles.
Rhode Island state statutes say that the Rhode Island DMV is authorized to take action against the license of a RI resident if they receive notice of an offense in another state. For example if RI is notified of an offense in another state that would cause your license to be suspended, if the violation had occurred in Rhode Island, than the RI DMV would suspend your driver's license.
The statute does not make it clear if Rhode Island will put all out of state citations on your RI driving record or assess points. We tried to contact the DMV but were unable to get through on the phone so you can try and check with your local RI Division of Motor vehicles to find out for certain if they will place this out of state citation on your Rhode Island driving record.
If the New Hampshire speeding ticket is placed on your Rhode Island driving record then your insurance company would see it the next time they pulled your motor vehicle record (MVR). It then would be up to your insurance provider's rating system to determine if this one traffic ticket would affect (raise) your rates.
As you probably know unlike most other states, Rhode Island does not monitor an individual's driving record through a point system. Instead, RI relies upon fines and the possibility of license suspensions or revocations to discourage drivers from being habitual offenders and continuing to receive traffic tickets.
The absence of a point system in RI means there is less tolerance for major offenses. Violations such as voluntary or involuntary vehicular manslaughter, driving under the influence of alcohol, and failure to stop in the event of an accident that causes injury or death result in mandatory license revocations lasting from 1 to 3 years.