I don't have a full answer here (at least not yet), but I do want to note that the kanji here are definitely not any kind of ateji - they are actually the exact opposite, a gikun (義訓 - 'meaning reading'), since 山 has no reading わさ, and 葵 has no reading び.
That means the etymology of the word わさび itself is unrelated to the etymology of the kanji わさび, and both should be treated separately.
The kanji chosen for this word means means 'Mountain Aoi (葵)'. Now, I'm not expert in Japanese botany (or in botany at all) - the best you can get out of me is probably recognizing a rose, and even that is only when I get pricked by one of it's thorns. :) But what I do know is that the word 葵 is used for several kinds of flowers that are not all of the same family. Some of them belong to the Malvaceae family, which also includes Genus Alcea (the flowers called Hollyhock in English), but some of them belong to an entirely different family. Wasabi is apparently similar enough to some of them, like カンアオイ, so the kanji chosen to represent it in meaning was mountain Aoi.
As for the origin of the word itself (disregarding the kanji its written in), there is no clear answer. It appears as early as the 10th century in a medicinal herbs manual as 和佐比, which is just an ateji rendering of the word, so it doesn't tell us anything about its origin. Some unproven theories can be found at Gogenjiten:
- It's an abbreviation of the expression 悪障疼（*わ*る *さ*わり ひ*び*く), which probably means something along the lines of 'causing horribly irritating pain'.
- The わさ in わさび comes from わしる, which is an old pronunciation of the verb 走る（はしる), to run, and it describes the pungent taste that "runs" all the way to your nose, while the び is the denasalized form of 実（み）, fruit or nut.
- The last theory is that the name comes from 早葵 , which was pronunced
wasaapupi in Old Japanese and written わさあふひ in historical kana usage (today it would be just わさあおい). This transformed (with a little elision and voicing of the ending syllable