True Story: Trump visited George Washington's home in Mount Vernon, Virginia. During the tour, he said if it was him, he would have put his name on the building, because that is the way people remember you.
The curator escorting him around the properly discreetly mentioned that Washington had his name on the nation's capital. He could have added and on an entire state.
He is having his name put on the check for one reason only, to make people who get the checks feel beholden to him and presumably more likely to vote for him. This is a classic dictator stunt.
Time will tell if it helps him at the voting booth, but I think it is more is likely to backfire, because it will remind people once again that he thinks everything is all about him.