An interesting question. A quick search of the Internet turns up:
en.wikipedia.org
Many of these, if not most, seem to have been given names by people who were not their original owner or creator, and may have served to punch up a story or legend about a famous figure. The only North American sword on this list is The Sword of State of South Carolina, which doesn't really fit the bill of a named sword.
Usually what shows up when you search for a named sword or sword with a name during the American Civil War is a sword that an owner has scratched or stamped his name or initials on or a presentation sword with the name of the recipient stamped, engraved, or etched on the sword or its scabbard. The practice of naming a weapon did exist in America, for example David Crockett had Betsy, Pretty Betsy, and Old Betsy. Daniel Boon had Tick-Licker, Simon Kenton had Jacob, Buffalo Bill had Lucretia Borgia and there were others.
Being the lore-besotted romantic that I am, you’d think that I would name all my rifles. Nope. Never have.
frontierpartisans.com
So, while my brief research didn't turn up any named swords, it did turn up named rifles. Perhaps there is something of substance in that. A new weapon for new mythic heroes? A break from Old World traditions? Obsolete symbols of power left in the past? Old Wrist Breaker was not the name of a particular sword but maybe it started out that way. Maybe further research will turn up individual names for American Civil War swords, but they will probably be for unsung heroes. If they were the swords of the heroes that we know, we would already know them.
Disclaimer: I didn't really search that hard, so if someone pops up with a name for General Forrest's sword(s) or the sword of Heros von Borcke or someone else I will not be surprised, but I suspect more soldiers North and South, high or low rank, named their firearms than named their swords.