Who chooses the names for naval ships?

It depends on the country, here are the US Navy standards.

I believe there is a navy committee that picks a list of actual names to use. This list is presented to one of the undersecretaries of defense who then either makes the actual selection or passes it up to his or her boss.

Your elected congress critters could probably put in a good word for USS Seelix.

The one who pays for them

Th classes of American Naval ships have different types of names—for instance the battleships are named after states. One of the names will be the name of the class—e.g. Iowa-class battleships. Smaller ships are named after smaller entities—cruisers were named after cities, but with the end of battleships, cruisers began to be named after states. Destroyers, still smaller ships, are named after American heroes.

Aircraft carriers [the largest ships afloat now] are named after presidents, implying named after the whole country, while reserving plenty of names.

And I am sure that Congressional politics play a part in which name is the class name, and which ships are named in that class.

Some other ship types are detailed here

This is the fundamental reference for American naval fighting ships

Smaller ships, like inshore fire support ships, are named after rivers, such as USS White River [LSMR-536 Inshore Fire Support Ship].

Sailors have their own nicknames for ships—White River was “White Rat”. A funny story from a Navy friend:

Sometimes ships have to tie up alongside one another if there are not enough berths at the pier. When the captain of the outermost ship wishes to cross over the other ships he must be announced by the ship name [as if he were the ship and the whole ship’s complement]. The captain of the outermost ship, White River, had to cross over my friend’s ship. Somehow the call was announced as “White Rat, crossing”
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Thanks, all! And cute story, @anartist :)

You’ve got to admire the occasional flash of humor displayed in some ship names. For example, the convention that US Ammunition Carriers are named after active volcanoes.

My uncle served aboard USS Vesuvius in WWII, Pacific Theater. These ships were never placed in forward positions, but even so, when they were asea they were always well protected at the center of convoys. The most dangerous part of the war for them was the night the war ended.

You have to realize that they often did duty while underway, tied between two larger warships and passing large volumes of high explosive to both sides simultaneously, all the while being banged and slammed by each of the two bigger ships, depending on sea conditions, and still in danger of submarine or aerial attack. Still, the most dangerous day of the war for them was VJ-Day, when they were anchored in a remote section of Leyte Bay in the Philippines. That night there was a general celebration by all of the hundreds of ships in the harbor… firing off celebratory shots in the general direction of the ‘deserted’ part of the bay where Vesuvius lay at anchor with all her hatches open. My uncle told me that the Captain was white as a ghost and in a near-panic from the first shot until the crew had all hatches closed and was starting to make way for the open sea to get away from the danger of this “friendly fire”.

cruisers were named after cities, but with the end of battleships, cruisers began to be named after states

Almost all US submarines are named after states, too.

For awhile the American boomers were being named after states, but the fast attack subs were named in other ways (such as the relatively recent Los Angeles class). Nowadays, I think the US is only building boomers (Virginia class, after the Ohio class). I’m not sure what kind of boat Jimmy Carter is.

The Virginias are attack submarines. They carry torpedoes and cruise missiles, not ballistic missiles. There are no US ballistic missile subs in the works.

USS Jimmy Carter (SSN-23), the third and last Seawolf-class submarine