The expression Noughties is formed from a derivation of the word nought (meaning 'zero' or 'nothing'), on the model of established references to other decades such as eighties, thirties, fifties etc.
At the advent of the 21st century and the third millennium, there was a whole host of ideas as to how we should collectively refer to the years beginning with zero. Among the suggestions were the Ohs, the Oh-Ohs, the Zeros, the Two-thousands, the Ones, the Nulls and, interestingly, the Naughties. Though in the US the word naught is a common spelling variant for nought, its usage is archaic in British English, which may explain why Noughties was favoured over Naughties.
The ultimate success of the noughties relative to the other candidates may partly be attributable to its catchy phonetic overlap with naughty, originally though to be a disadvantage, but creating the potential for wordplay as in naughty noughties or noughtiness.