His relationship to Beatrice is immediately established as one of a familiarity. If Beatrice had just referred to Benedick by his real name, than the audience would have no understanding of the level of knowledge they have of each other. This nickname lets us know that they not only know one another, but that there is some history invovled between the two. The first scene between the two of them will develop this back history further, by letting the audience know that Beatrice "knows Benedick of old." They could be old friends (their teasing is still too light to suggest they are enemies) or - as we learn later - a former romantic couple.
"Signior Mountanto" is what Beatrice calls Benedick when he arrives. This is her way of saying that Benedick is a ladies' man (see the discussion of "Knowing Aforehand," linked below). Beatrice and Benedick seem to enjoy insulting each other, but they are insults that soon turn into endearments when they realize they are in love.
Signior Mountanto is Benedick, and Beatrice calls him "Signior Mountanto" in Act 1, Scene 1. This would have been seen as humorous by an Elizabethan audience, as mountanto refers to "montanto," which is a term associated in fencing to an upward thrust. Beatrice is implying, therefore, that Benedick has bad fencing skills. The technique used here is a pun.
Signor Montanto is Benedick. she refers to him as thins because there is a 'merry war' between them and she calls him this to tease him