I've traveled to Paris twice by myself so far, and while I love going with my husband, there is a true freedom to traveling solo. As others have said, Paris is a wonderful city to be by yourself. You can do what you want when you want.
I would suggest hanging out in the gardens! I love the Tuileries, Luxembourg, the Jardin des Plantes and Parc Monceau. All are wonderful and unique. If you go to Parc Monceau, definitely check out the Musee Nissim de Camondo. Another great place to hang out is the small park behind Notre Dame (and it has free bathrooms if you need them!)
This may sound corny, but just let Paris envelope you. Have a loose plan for the day, but listen to your gut and if you'd rather sit in a cafe rather than go to a museum, that's fine! Just BE in Paris.
You'll have a great time!