That's a joke at my expense, as in And after all that discussion they didn't show up—that's one on me. This phrase must be distinguished from that's a new one on me, which means “this is the first time I've heard of or seen that” (as in A checkerboard rug—that's a new one on me). Both idioms can be used with other personal pronouns (for example, that's one on you) and date from the early 1900s.
The American Heritage® Idioms Dictionary Copyright © 2002, 2001, 1995 by Houghton Mifflin Harcourt Publishing Company. Published by Houghton Mifflin Harcourt Publishing Company.