It's kind of like the "there's no such thing as bad publicity" rule. All they care about is making sure you see their name/URL often enough that you'll think of it if you ever do decide you want to buy gil from someone.
So, they're trying to stay off of people's blacklists so they can keep spamming them. They've been using the /tell and then disconnect pattern for a while, to exploit SE's rule that you cannot blacklist someone unless they're online (which is another rule I wish SE would change to avoid this exploit). But a lot of players are getting pretty fast about blacklisting, and can get it in before the gilseller has time to disconnect. By adding in the extra Friend invite, they make it take a moment longer before anyone can blacklist them, and that gives them the time to logout, so they can't be blacklisted.
They're probably also hoping to catch an occasional new player who's just joined the game, is looking for friends to play it with, and will accept a random friend invite mistaking it for a genuine offer of friendship. Then the gilseller can't be blacklisted for a much longer time (until the player realizes what's going on and finds where to remove them from the friend list), and they get the new tool of gil spam through moogle-mail.
