N.H. Rev. Stat. § 655:37
If, after the holding of a state primary election, a vacancy exists for any office on a party ticket, such vacancy may be filled as provided in this section. The appropriate party committee shall notify the secretary of state in writing of a person they designate to fill the vacancy. The person so designated shall, no later than the first Tuesday following the primary election, file with the secretary of state a declaration of candidacy as provided in RSA 655:17 with the understanding that, where the form says primary election, it shall be construed to mean general election. If the candidate is designated for the office of governor, executive councilor, state senator, or state representative, he or she shall also file, no later than the first Tuesday following the primary, the appropriate affidavit as provided in RSA 655:29. Any candidate who has not filed all the forms required by this section within the required period of time shall not have his or her name printed on the state general election ballot for that office.
RSA 655:37
1979, 436:1. 1983, 426:14. 1994, 4:7. 2010, 317:10. 2012, 113:5, eff. May 31, 2012.