What placename should we use for the New Hampshire Grants in the early 1770s

Question

Province of New Hampshire? Province of New York? Or use Vermont? Vermont Republic? I'm sure the Green Mountain Boys would have an opinion...

Answer 1

a few possibly informative sites for you 

New Hampshire Land and Property Genealogy - FamilySearch Wiki

Map, Available Online, 1700 to 1799, Land Grants | Library of Congress

4 Early Land Divisions – New York Period | Brattleboro History

Old Historical City, County and State Maps of New Hampshire

Answer 2

The land grants, totaling about 135 (including 131 towns), were made on land claimed by New Hampshire west of the Connecticut River, territory that was also claimed by the Province of New York. The resulting dispute led to the eventual establishment of the Vermont Republic, which later became the U.S. state of Vermont.

but it gets tricky because much of Vermont was part of the very large Charlotte County New York for some years.   It sometimes is necessary to do a genealogy of the area you are studying.  OLD MAPS may be helpful.  

Then where were the courthouses?  Because of harsh winters some counties had two Court houses.  Sometimes those records have been moved to Albany.    Good luck in your search!  It is a labor of love.