The phrase sounds entirely suited to Tudor yokels and is a stock in trade of any author wishing for a shortcut to convey a sense of 'Olde Engylande'.
A bodkin is a small tool for piecing holes in leather etc. This term borrows the early bodikin version of that word, not for its meaning but just because of the alliteration with body, to make a euphemistic version of the oath God's body. This would otherwise have been unacceptable to a pious audience. That is, odds bodkins is a minced oath.
Shakespeare ignored the impropriety in Henry IV Part II, 1597:
God's body! the turkeys in my pannier are quite starved.
Perhaps he was rebuked for that. In any event by 1600, when he wrote Hamlet, he had gone halfway towards the euphemistic version:
My lord, I will use them according to their desert.
God's bodykins, man, much better: use every man after his desert, and who should 'scape whipping?
The first to use 'odds bodkins' in something approaching the current spelling was Henry Fielding, in Don Quixote in England, 1734:
"Odsbodlikins... you have a strange sort of a taste."