The other "H" is George Hamlin, Howard Hamlin's father.
This is first discussed in Season 2 Episode 8, "Fifi". As per Howard Hamlin's dialogue:
Back in the day, right out of law school, I thought long and hard about hanging up my own shingle.
Oh, I was ready to take on the world, make a difference.
Dad talked me out of it.
Wanted me to join the ranks here, throw another "H" in the firm's
In Season 3 Episode 5, "Chicanery", Howard Hamlin, while testifying under oath as a witness, was asked directly who the other Hamlin is in HHM. He again mentions that it is his father.
Howard Hamlin's father's name is mentioned in Season 4 Episode 1, "Smoke":
Charles joined George Hamlin's solo practice on Central Avenue
in downtown Albuquerque. Over the course of 23 years, the two men
built what became known as Hamlin Hamlin McGill into a nationally-recognized firm.