The story
Tartle is Scots, and the Scottish National Dictionary gives it a precise meaning: to hesitate, to be uncertain, especially when trying to recognise a person or a thing. An example recorded by John Jamieson in 1808 has someone saying that they tartled at a man because they could not be sure who he was. The word could also be used of a horse that balks, or of a buyer who hesitates over a bargain. Allan Ramsay's collection of Scots proverbs from 1736 has an empty purse making a tartling merchant.
It worked the other way too. To tartle on someone was to recognise them after all, once the doubt had passed. And as a noun, a tartle was the hesitation itself. Where the word came from is uncertain; one suggestion is an Old English verb meaning to totter or waver, but the dictionary admits the evidence is missing.
Online the word now carries a narrower gloss: you are introducing someone and their name will not come. The Scots dictionaries do not record that scene; it is a recent narrowing of the old sense, not a separate meaning, and it fits. The tartle is the half-second when the face is familiar and the label has gone.
English has blanking, and everyone knows the stall. Scots simply gives it a word that sounds like what it is: a small trip in the conversation.
There is also an unrelated tartle in Scots, for a matted tuft of wool at a sheep's tail.