I know that the Scottish prefix 'Fitz' means 'illegitimate child of', but I'd like to know if there's an Irish language equivalent? If there is, what would the Irish version of 'FitzWilliam' be?

Fitz IS an Irish prefix, and though I'm no expert on name origins, I've never herd it used in a Scottish context. I do know, or at least am reasonably certain, that Fitz is derived from the the Anglo-Norman pronunciation of the French word for son "fils." Hence, when the Anglo-Normans arrived in Ireland, they adopted Irish naming conventions such as "the son of so and so" or in this case "the son of william" which was rendered in Anglo-Norman French as "Fitzwilliam." I'm also pretty certain that there is no allusion to illegitimacy in the use of Fitz. Again, it simply means the son of.

Why, are you a bastard called William?