Spiders are known as Arachnids which leads one to the guess of fear of spiders. This is from the Greek legend of Arachnae, who was turned into a spider for telling the truth.
But in fact, the "arach" part does indeed refer to the peanut, or groundnut (Arachis hypogaea) The best I can find out about that is
"Arachis was the name, among the ancient Greeks, of some leguminous weed, not the peanut. The precise ancient meaning of arachis is unknown but the components of the word suggest that arachis may have been composed of:
/a/ Greek ‘not’ + rachis Greek ‘spine, backbone’
so that the flower of this ancient weed perhaps appeared to have no central support structure or axis or rib. In 18th-century Botanical Latin, rachis was the part name for the main axis of a leaf or the rib of a frond or leaf. Rachis is still used in botany with this meaning — but we don’t know if the ancient Greeks used it in that sense. Such a derivation is etymologically possible. But it’s just a guess, kiddies — although a well-educated guess — he said humbly. "
And ... butyro-, butyr- +. (Greek > Latin: butter; from bou[s] , "ox, cow" plus tyro[s], "cheese").
So, accurately, it should refer simply to fear of peanut butter. The sticking to the roof of the mouth part is ... dare I say it ... tongue in cheek, I believe!!
Watch for this question in your next trivia game! It could be good for a luggage tag.