Ad nauseum is a common misspelling of ad nauseam, even among careful writers. It’s a subtle enough error to slip past the copy editors of reputable publications (recently here, for example, until a commenter pointed it out). The wayward spelling ad nausea also appears, sometimes as a joke. I mentioned ad nauseam before, along with a mnemonic to commit the correct form to memory, but the prevalence of ad nauseum suggests the need for a dedicated post; some background information might help.
Ad nauseam is an adverbial phrase, clearly Latin: ad here means to; nauseam is nausea in the accusative. So ad nauseam means to (sea-)sickness, i.e., to a nauseating or sickening extent, often figurative for “to a very tiresome or boring degree”. To do anything ad nauseam is to be sick of doing it. The phrase is said to originate from argumentum ad nauseam, a term in logic similar in sense to argumentum ad infinitum. It has been used in English since the 17C.; there’s a slightly earlier form, usque ad nauseam or ad nauseam usque (all the way to nausea, i.e., to the point of nausea), which is rarely encountered nowadays.
The difficulty with spelling ad nauseam probably results partly from its Latin origin: we recognise its Latin-ness but associate the language more with -um endings. There’s also the indistinct pronunciation: though it’s correctly pronounced /ad ‘nɔːzɪam, -sɪam/, the closing /-am/ is often rendered as /-ɘm/. The unstressed vowel sound ɘ is a schwa, known in Spelling Bee circles as “the dreaded schwa” because it’s so difficult to guess which vowel lies behind it. Thus it is with ad nauseam. To remember the spelling, think of nausea. Or more elaborately, say to yourself: “I am nauseated not to know.” If you think it might be -um, consider ummm a hint that you should keep wondering about it another moment, until you think: “Ah! Now I am sure: it’s ad nauseam.”
Google search hits shouldn’t be taken literally, but they can offer a crude indication of popularity; and a comparison today shows the erroneous form topping the traditional by 530,000 to 476,000. I could say I see ad nauseum ad nauseam, but that would be an overstatement. (Not that there’s anything inherently wrong with exaggeration.) It’s possible that ad nauseum will eventually be accepted as standard — English is not Latin, after all — but for now I advise the use of the standard spelling ad nauseam.