I'd advise Duran against renaming the sun because it already has good branding and market share. But she needs to trademark the word "sun." She could also assert copyright over sunset, sunrise, and solar eclipse photos. She could insist that the credit line read: "Sun, courtesy of A. Duran."
To get people on board, Duran will need to launch a multimillion dollar add campaign, with a catchy phrase like "Got Sun?" And if solar power use declines, she can advertise "New Sun."
What happens if you don't pay the solar power bill? Duran can't unplug the sun. But for an investment of a few $10 of billions she could launch a lot of sunshades into geosynchronous Earth orbit. They would be positioned to cast little shadows on the homes of people behind in their solar power payments. Imagine walking down a neighborhood and seeing a house in permanent shadow. Oh the humiliation. "Mommy, I'm going over to play at Joey's, he's got sun."