Dr. Z found a link that identified what initial letters went with which counties on the NH plates from 1949-1975. A further interesting bit is that one can see how it progressed.
Plates originating in each county took the first initial of the county, at least to start. Where two or more counties start with the same letter, one got precedent for whatever reason, and the others took a letter further on in the name: E for Cheshire (H was already taken by Hillsborough), O for Coos*. As each county issued more plates, it moved to the next letter in the alphabet. Hillsborough was and is far and away the most populous, and likely moved to the "I" pretty quickly. This rather put Cheshire up against a wall, as all its letters were already taken by another county, so when it overflowed, it got the Z. Coos, with only three distinct letters and two of those taken, would have had to get something similar - something found in no other county's name - had it ever filled up. But despite being the largest geographically, it's pretty empty. Still, pretty poor planning to give the C to Carroll, which still had the A, O, and L available.
Rockingham exhausted the R before Hillsborough had used up J, and so got the K. Hillsborough needed another one eventually and took the L. I don't know where they would have gone after that. Rockingham eventually needed another - little surprise, as that went from a smallish population to one of the larger - so they went backward and grabbed the P.
You can see that the system was about to fill up. A, Q, U, V, W, and X were all that were left. A would be ambiguous, being in so many counties, and U would likely go to Hillsborough or Sullivan. Not much left for everyone else.