What is the etymology behind したっけ anime shows the children saying ! translated as 'See ya!' and other terms, these are from Hokkaido dialect. I guess you can switch to other forms to say 'See ya': Anyway, there's any relation with ? What is the etymology resulted in ? The way that sounds, it's like an affirmation than a question.

No, this phrase isn't cognate with Standard Japanese .

literally means what in Standard Japanese . The demonstrative is omitted because the whole context before is considered to stand in place of it (colloquial omission of this is also common in Tokyo). The part shares the same origin with Standard ("(what) again?"), that is Classical indirect past , but has diverged from it to mean "(after it) then".

Thus, Hokkaido-ites say "Now then!" to mean "See you!", but this is exactly parallel to Standard Japanese too, where literally means ("so") + ("if be -- then") in older way of speaking.

**EDIT**
is seemingly used in the same meaning in Ibaraki dialect, too. Hokkaido dialect is a mish-mash of various Honshu dialects, so it's possible that this part of grammar was exported from Northern Kanto.