Great question! This of course comes just after the stranger makes his presence evident to Tom, when Tom is in the forest. Note what the story tells us about the trees:
Tom looked in the direction that the stranger pointed, and beheld one of the great trees, fair and flourishing without, but rotten at the core, and saw that it had been nearly hewn through, so that the first high wind was likely to blow it down. On the bark of the tree was scored the name of Deacon Peabody, an eminent man, who had waxed wealthy by driving shrewd bargains with the Indians. He now looked around, and found most of the tall trees marked with the name of some great man of the colony, and all more or less scored by the axe. The one on which he had been seated, and which had evidently just been hewn down, bore the name of Crowninshield; and he recollected a mighty rich man of that name, who made a vulgar display of wealth, which it was whispered he had acquired by buccaneering.
Clearly, the tree that is rotten at the core can be shown to symbolise a man who outwardly appears to be great but it rotten inside. Likewise, we can infer from the men's names being hacked into the trees and the black man carrying the axe that man himself has hacked the names into the trees, as if claiming the men's souls for himself. All those who are "great" in the eyes of society are shown to have links with the black man, and are shown to have gained their wealth through illicit or questionable means.
The names can imply a number of things: wealth is obtained in base manners; leaders in the community are corrupt; the Devil can destroy as easily as he gives (the trees can be cut/souls can be harvested); etc.
By showing the names, the Devil is also using a sly trick of persuasion - after all, if these "upstanding citizens" have made deals, then who is Tom Walker to refuse the opportunity as well (for it must be worth the risk in the long run)?