Well, Ed's kicked off several more trains of thought. (Shame on you, Ed.)
Paul Silverstone's The Sailing Navy, 1775-1854 does list two ninetenth-century USN vessels named Boxer: a brig launched in 1815 (and presumably named after the Enterprise's victim) and a schooner launched in 1831. The name was also used, briefly, for a blockade runner that was captured and taken into the navy during the Civil War. Maybe the most famous earlier U.S.S. Boxer was the steel-hulled (I think) brigantine that was purpose-built in 1905 as a training vessel for the midshipmen at Annapolis. At any rate, the DANFS describes the carrier Boxer as the fifth warship of the name.
Off the top of my head I can think of three other U.S.N ships named after other warships that didn't serve under the American flag: the frigates Guerriere and Java (both named in honor of British frigates sunk by the Constitution during the War of 1812) and the sloop of war Peacock (named after a British sloop sunk by the Wasp). In those days, of course, the use of the names of vanquished enemy ships was a common practice among several nations.
There were in fact two sloops named U.S.S. Peacock. The first was built during the War of 1812 and broken up in 1828. The second was launched the same year (having been funded in part with money appropriated for "rebuilding" the old one - a common, and perfectly legal practice), and gained some fame as a participant in the "Wilkes Exploring Expedition" of 1838. She was wrecked off the Columbia River in 1841, with no loss of life. I'm not sure whether the Navy ever used the name again; I rather suspect it wouldn't be considered a good idea today.
The name Franklin is an old one for American warships. Several of the schooners informally "commissioned" by George Washington during the 1775-1776 siege of Boston were named after prominent people in the independence movement: Hancock, Franklin, Warren, Lee, and Washington. The name Franklin was also used for a captured Tripolitan merchantman used as a Navy supply ship during the Barbary Wars, a ship-of-the-line commissioned in 1815, and a big steam frigate commissioned in 1864. (I'm getting all this stuff out of Mr. Silverstone's book, with a little help from the online version of the DANFS.) So the carrier Franklin was, officially, the fifth USN ship to bear the name.
The question of naming warships after Civil War battles is interesting. Apparently at least a couple of Union vessels were given the names of recent battles (Gettysburg and Antietam) while the war was still going on. After 1865 the practice seems to have ceased - presumably for reasons of Southern sensibility. This is the sort of thing that's hard to look up systematically, but on the basis of a quick search through conveniently-available websites the first post-Civil War ship I could find that was named for a Civil War battle was the Essex-class carrier U.S.S. Antietam (CV-36), launched in 1944. Since then, six of the big new Aegis-type cruisers have been named for Civil War battles: Mobile Bay, Antietam, Gettysburg, Shiloh, Vicksburg, and Chancellorsville. (One could, I suppose, argue that several of those are named after cities. If one wanted to be really perverse, for that matter, one could argue that the various ships named Yorktown were named after a city - or after a Civil War battle, since there was some fighting near Yorktown during McClellan's Peninsula campaign. But one can carry this sort of thing too far.)
If Mr. Trebek and his consultants ever do get enmeshed in this topic, they're in for trouble.
Later edit: Russ and I apparently were typing at the same time, and coming up with some of the same stuff. We really ought to have better things to do on a Sunday morning. Russ - I do hope you aren't in the path of Hurricane Gustav, which looks mighty ominous.