Let's see...
Elle's tiny little Miranda wasn't named by her, but by its late previous captain, and was the USS Proud Moth because it was small, lightly armed, but still did its best.
Once she was able to name her own ships, she was made captain of the USS Richard Feynman, an Intrepid-Class ship... this poor vessel was destroyed in an altercation with the Borg, but a later Intrepid-R was commissioned to the same specs and christened the USS Richard Feynman II.
She currently flies the USS Into the Unknown, a Nebula-class science vessel.
She entered Starfleet to serve as a medical officer in person, and to discover new types of spatial phenomena as a captain, and her scientific bent is I hope apparent from her ship names ;)
She has recently been considering putting in a request for a transfer to one of the newly-developed D'Kyr Vulcan Science Vessels that Starfleet has been talking about in classified communiques, but she hasn't yet decided on an appropriate name.