I’ve seen a lot of people who think that the Blackbeard Pirates will just kill Garp. However, I think the story has already told us what the Blackbeard Pirates plan on doing with Garp the hero.
In chapter 1080, we see the reason that Koby was captured by the Blackbeard Pirates, and that was to be used as a bargaining chip to make Hachinosu a true nation, one that is a part of the World Government.
However, the biggest problem in Blackbeard’s plan was quickly pointed out by Kuzan, Koby is a member of SWORD and thus the marines can cut all ties with him.
Regardless Blackbeard wants to go through with it, but obviously with the most recent chapters we can see that this plan didn’t work out. However, he has an even better replacement. Someone with the hero name who doesn’t seem to be a member of SWORD at all, Garp.
The marines have to act on Garp, he is a legendary figure and a vice admiral and I think with the Egghead incident it’s very likely the government will cave in to his demands, igniting whatever Blackbeard plans to do now that he is a nation. Whether it’s sneaking someone in with the holy tribute or possibly seeking a meeting with the Gorosei, there has to be some reason Blackbeard wants this outcome and it will be relevant after Egghead I believe.