About his style of governance, nothing. Trump is too much of a crazed autocrat to permit anyone in his administration to tell him what to do. If you don’t believe me, just ask Mike Pence.
But it will tell us where, if anywhere, he thinks his vulnerabilities are in the coming election. If he picks Rubio, he’s concerned about keeping the Haley voters on board. If Vance is the choice, he wants to fire up the base, as usual. If he chooses a woman, he’s still worried about how he is perceived by over half the electorate.
If it’s Burgum, he doesn’t think he needs any help to win the election; he just doesn’t want someone who can ever outshine or criticize him.