Thanks for flagging this.
Looks like she's shown as "Thomas J Slape" or "Tom J Slape" on the 1860, 1880, and 1900 U.S. Federal Censuses.
I think I see where the connection might be coming from...he's listed as "Tom Slape" on his son James Lewis Slape's death certificate, but that record has been incorrectly indexed as "Jam. Slape." Based on that record, people are probably assuming that he was a middle-namer and that his son was named for him. I totally get why they'd think that!
Here's a mention of a family Bible having him as "Thomas Jefferson Slape."
If the manager doesn't have a source for "James," I'm happy to edit it to show "J.," with a note explaining the matter and how there are multiple theories.