The variants as right as ninepence and as neat as ninepence are just as common as the 'nice' version (also, less commonly, 'as clean as ninepence' and 'as grand as ninepence') and it isn't clear which came first.
There are suggestions that this expression derives from from 'as nice as nine pins'. In the game of Ninepins (Skittles) the pins are set out in a square. For the game to be fair this must be done neatly and accurately or, in the old parlance, nicely. That derivation is attractive but seems unlikely, as there is no early records of 'as nice as nine pins' in print. We might expect that form to exist if the 'ninepence' version derived from it. The 'ninepins' form, in the guise of 'as smart as ninepins' isn't found until the 20th century and it is reasonable to assume that is a simple mishearing of the earlier 'as neat/clean/grand as ninepence' versions.
The earliest known recorded form of the phrase is 'as neat as ninepence'. The first citation of that is in James Howell's English Proverbs, 1659:
"As fine as fippence, as neat as nine pence."
The 'fippence' (five pence) in that makes it clear that the reference is to money rather than to skittles. For it to appear in a list of sayings viewed as proverbial it must have been in existence for some time before 1659. There was a ninepence coin in circulation in the 16th and 17th centuries, although there was nothing especially neat or nice about it. The rhyming style of that citation suggests that the 'neat' and 'nice' were chosen just for that reason.
The 'as nice as ninepence' form didn't arise until recently - late 20th century.