New Hampshire has Election Day Registration. 

Qualified individuals may register to vote at the polling place on election day at all elections.  You will be asked to show proof of identify and domicile and, by signing the registration form, verify you are a citizen and old enough to vote.

Photo ID 

• Driver's license (New Hampshire or out-of-state)
• a U.S. or state-issued photo ID
• a U. S. passport, or any other proof determined reasonable by the supervisors of the checklist.
• It must have your address on it.  If not, bring along one of the nonphoto IDs with your current address.

Nonphoto ID

• Current copy of a utility bill, bank statement, government issued document or other proof deemed acceptable by the supervisors of the check list.

• If you do not have proper ID, New Hampshire gives you the option of signing an affidavit attesting to your qualification as an eligible voter - age, citizenship, address in the town or city you are voting in. 

Check with the town or city clerk in the municipality where you plan to register to determine if that town/city has adopted a list of alternative documents that will be accepted as presumptive proof of domicile. Some towns accept forms issued by your college or university.