Because the United States had just been dragged into a two ocean war and only had seven active carriers. The next fleet carrier to come off the slips was the USS Essex which did not commission until almost a year later in December 1942. The United States needed every heavy combatant they could deploy as soon as possible. The next US battleship to be commissioned was the USS South Dakota which was commissioned in March 1942. The North Carolina class, launched only a few months before Pearl Harbor was only two ships strong. Between Pearl Harbor and the launching of the USS Essex the United States lost four fleet carriers, the USS Lexington, Yorktown, Wasp, and Hornet. In the same time frame the USS Saratoga was torpedoed twice and the USS Enterprise was bombed.

Despite these losses and the long gap between entry into the war and deployment of new build warships the US still had to fight a war. At Pearl Harbor were a large number of damaged, but still combat worthy warships if the US could get them into the war. Many of them ended up as the USNs heavy combatants in the Aleutian campaign.

It just seems a odd use of limited resources to repair, refloat, and refurbish obsolete 25 year old battleships for a war dominated by the aircraft carrier.

Maybe looking back from a modern perspective it looks that way. But remember, the dominance of the carrier was established during the war, not beforehand. There was still pressure to design, build, and employ battleships up to and during the war. The US was still planning to build the Montana class until the lessons of the war showed that additional Essex class carriers would be more useful. Second, remember the limited number of carriers available. They could not be everywhere at once and when they could not battleships continued to find use. The Aleutians, which I mentioned before was one place where they were used, the Mediterranean where the British kept losing their carriers and the Italians didn’t have any was another. In the USN the older battleships were employed heavily to bombard islands in the Pacific and support the Marines. In the Atlantic they were used as convoy escorts against the threat of German surface raiders until the allies were ready to attack the German shore fortifications. The fast battleship classes (South Dakota, North Carolina, Missouri) provided heavy escorts to USN carriers in the Pacific. In the Atlantic they were part of Operation Torch and helped neutralize the Vichy French Jean Bart and her defenders.

To sum up, the USN repaired their battleships because they were still useful even if they were no longer serving as the fleet’s main striking arm and they were already built. They could not be deployed directly against Japanese carriers or air bases, but they still offered heavy firepower in locations where those were not a threat or American carriers were not available.

Firstly, some of the battleships weren't heavily damaged. The USS Tennessee, USS Maryland and USS Pennsylvania took some bomb damage during the attack. While this damage was significant enough to prevent them being ready for service immediately, all three would be in service by February 1942. Writing off these ships because of minor damage made little sense.

Secondly, the battleship was not completely obsolete in 1941. While in hindsight we know that the battleship had ceded it's primary role to the aircraft carrier, this was not obvious in 1941. At the time, the primary example for the role of the carrier in modern warfare was the Royal Navy, in the Mediterranean and North Sea. In both of these theatres, carriers had proved useful adjuncts to the battlefleet, which had proved to be the decisive tool. At Matapan and during the Hunt for the Bismarck, the RN's carriers had damaged Axis ships, allowing the battleships to destroy them. In several other battles, chiefly Calabria and Cape Spartivento, the carriers had been useful for scouting, but had done little damage to the opposing fleet.The loss of the Glorious demonstrated the vulnerability of carriers to battleships. While these examples were mainly due to British doctrine, which emphasised the carrier's role as a subsidiary to the battlefleet, they were the only example of wartime carrier use available pre-war.

Thirdly, the resources needed to repair and refloat these ships weren't needed to produce carriers. The primary bottleneck on carrier production was slipways. This put a hard limit on carrier production - there were only so many in the country that could launch a full fleet carrier. Any resources assigned to carrier production over this would have been wasted. Raising the battleships used salvage engineers, which weren't in high demand for carrier production. Refurbishing and repairing them were done in drydocks on the West Coast, which weren't needed for the carriers.

Finally, the Navy needed all the ships it could get it's hands on. While the Japanese had no plans to attack the West Coast, the Navy was paranoid about such an occurrence. Such paranoia was boosted by the rapid pace of the Japanese advance until the battles of Coral Sea and Midway. The three lightly damaged battleships referred to above were held on the West Coast, just in case. Refusing to refloat three of it's ships, especially those perceived by the public to be it's most powerful, would have been political suicide. The battleships raised at Pearl Harbor would go on to find productive employment bombarding Japanese-held islands. More modern battleships were needed to escort the carriers, and couldn't be spared for such duty. They would also prove that they could still fight, at the Battle of the Surigao Strait in 1944.

The US needed all the firepower it could get after the attack, and it was cheaper and faster to repair some battleships that got the job done fine rather than build brand new ones.

Although the war would come to be won by aircraft carriers in the pacific, the "battleship being obsolete" line of thought had not completely taken the forefront of naval military thought yet.