This seems to have been a variation of spelling that developed in Middle English.
The word derives from Old French, the Old French Spelling was "arguement", with an "e". That was derived from the Latin "argumentum", (argu- (make clear) + -mentum). Note the Latin doesn't have an "e"
The French "e" came to be inserted as the related verb ending is -ere in Latin and -er in French, (as "arguer") French tends to insert an "e" when forming "-ment" nouns from "-er" verbs. The "e" was then dropped in English, probably to better match the original Latin.
Words like "management", for example, are not based on any Latin original. They were formed in English from "manage"+"-ment", and haven't lost any letters.
The only rule is that there is no rule, but Latin can sometimes justify how a word came to be spelled.