I know they were all named for states (except for the USS Kearsarge), but how was the specific state chosen? Was it a lottery, or by population, or what?
This is an interesting question because it is both an entirely trivial topic and a very important subject. On the trivial side the name of a ship is meaningless in terms of it's capability and performance, and any designation would suffice for communication and record keeping. On the other hand capital ships are a massive investment, a tremendous national risk born usually of need, and a major source of national pride. In that context naming them takes on an almost mystical, and very real political import.
The Navy system for naming ships is a tradition born of law. That sounds a bit backwards, and it is, but there you have it. Congress gave the authority to name ships to the Secretary of the Navy in the early 19th century. The specific passage read, "That all ships of the navy of the United States, now building, or hereafter to be built, shall be named by the Secretary of the Navy, under the direction of the President of the United States" which is both very clear and somewhat ambiguous, because it clearly assigns ship naming to the Secretary of the Navy but specifically mentions the President as well. Since the Secretary of the Navy is assumed to work under the direction of the President already presumably this allows for the President to select names, and order the Secretary of the Navy to assign them to vessels. This law also called for ships of the first class to be named after states, and ships of the second and third class to be named after rivers and cities. So legally as of 1819 the Navy had an established system of naming ships, with either the Secretary of the Navy or President assigning names, and capital ships being named after states. But there were no battleships in 1819, and the US Navy was small and still a frigate Navy.
When steamships and monitors came along there were new revisions of the 1819 law, basically reiterating the part about Sec. Navy under President's direction and redefining the "first class" bit to reflect the evolution in ship design. When the first battleships came along the law was amended again, but in an interesting way. In 1898 a Naval Appropriations Act was passed by Congress calling for battleships and monitors to be named after states and added that they were not to be named after any city place or person until the names of the states had been exhausted. This law did not assign the duty of naming ships to any specific person though, unlike the three previous laws that had all spelled out Sec. Navy under the Presidents direction. At this point there was no legal process for selecting the name, or giving any individual or group authority for making these selections, but a legal system the names themselves had to comply with.
To make things even more complicated in 1908 Congress acted again, this time to exempt monitors from the provisions requiring them to be named after states and allowing them to be named by the President (no mention of the Secretary of the Navy). The US Code of 1925 maintained the ship naming rules as to naming battleships by state, and did not mention the Secretary of the Navy.
The current system is this. The Navy has multiple sources for recommending names based on various factors. The lists are compiled by the Naval Historical Center, which forwards the list to the Chief of Naval Operations. The Chief of Naval Operations submits suggestions to the Secretary of the Navy once a year, and the Secretary of the Navy makes a final decision on naming the ships. Some capital ships are given state names, Battleships were always given state names (except the Kearsarge) up to and through the final WW2 BB's that were built. I say up to and through because 7 additional WW2 battleships were ordered and cancelled. 2 of them, the Illinois and Kentucky would have completed the Iowa class. There were 5 planned Montana class battleships, these would have been larger and more powerful than the Iowa class and would have been named the Montana Ohio Maine New Hampshire and Louisiana.
Since WW2 we have stopped building Battleships and state names have gone to ballistic missile and attack submarines (the Ohio and Virginia class). These are class names selected by the Secretary of the Navy. As far as specific state names going to specific Battleships or other capital ships, the names are rotated, with the goal of every state name being used periodically, they are also rotated by geography, with states right next to each other not used at the same time.
From everything I've ever read on the subject, there doesn't seem to be any set process for which the Secretary of the Navy selected what particular state name to use to christen a new battleship other than his own judgement and what names were currently not used by actively commissioned ships. Though I would imagine from time to time, you'd have a congressman, dignitaries, or other political officials pushing for his or her state to be chosen as the namesake whenever a new ship was ordered or laid down.
Also, up until 1916, battleships weren't the only USN ships that had state names. Armored Cruisers and monitors also received state names up until that point, however, the Navy Department in 1916 quickly saw that a shortage of State's names available for ships would occur if this naming convention was continued into the future. There were only 48 States at the time and as of December 1916 there were 36 battleships, 3 monitors and 30 cruisers (though I'm not entire sure how many of those cruisers were armored at the moment so that number is probably a lot lower), so obviously something had to be done to avoid running out of names for new battleships.
So in 1916, the Navy Dept. made the decision to change the naming convention so that only battleships were named after states (with the exception of the USS Kearsarge which was laid down in 1898) as required by law and renamed all active non-battleships with state names after cities in respective states. This gave the USN plenty of names to choose from whenever a new battleship was ordered, as it was highly unlikely the USN would have more than 48 battleships commissioned at any one time. This seemed like a prudent thing to do, as 1920's naval doctrine was highly centered around the use of the battleship and the big gun fleet, but this obviously changed quite a bit once The Pacific War broke out.
After World War II, the use of the battleship continued for a few more decades (and for a brief period in the late Cold War), but the aircraft carrier (which has a completely different naming convention) took it's place as the center of the fleet in the coming decades as naval aviation quickly began to obsolete the big gun doctrine. As a result, no further battleships were constructed after the USS Iowa and USS Wisconsin were launched during WWII, and this is why some states did not receive battleships named for them. Though there were some Iowa class battleships in the initial construction stages (before being canceled) that were planned to be named after States that had not previously had a battleship named after them.