The Oxford English Dictionary says the origin is uncertain, so there's not much chance that any of the rest of us will be able to come up with an answer. Certainly, it's been in use since the early 1800s, though long gone now. Sorry!

Likely, "Some of the issues [sixpences] of George II were designed by John Sigismund Tanner, and the denomination was known as a tanner right up to decimalisation." http://www.tclayton.demon.co.uk/sixd.html

I like the idea of the coin's being named after the designer, Tanner, Answerbok, but the evidence is a bit thin. Tanner died in 1775 and George II died in 1760, but the very first use of the word in this sense did not appear in print before 1811. You would have expected the word to appear during their lifetimes rather than almost two generations later.

Here are three more possibilities, the first of which seems - to me at least - to be more likely. The Italian word for 'small change' is (was?) 'denaro', which sounds pretty like 'tanner'. There is also the Romany word 'tawno' meaning 'little one' and the sixpence was a small coin. Again, the word sounds very like 'tanner'. Finally, there may be a connection with 'thaler' - from which we get 'dollar' - an old German silver coin. Not quite so close in sound, but still a possibility.

All-in-all, I suspect it's best to go with the OED and say 'origin uncertain'. Cheers