You could write a story where she has adventures in time and space inspired by her love for the show.
Well, you could imagine her getting involved in a haunted hospital and using her knowledge of Doctor Who and Rose to solve the mystery. Or perhaps she has visions of Rose guiding her through difficult medical cases with a spooky twist.
You could focus on a character from one show ending up in the world of the other. Like if one of the Animaniacs gets lost and is found by the Doctor. It could lead to all sorts of interesting plot twists and character interactions.
You could write a time-travel adventure featuring Alexander Holmes and the Doctor facing off against a powerful alien threat.
You could write a story where the Doctor and the Muppets team up to save a magical world. It would be full of fun and adventure!
You could write a crossover story where the Doctor teams up with the Avengers to save the universe from a combined threat.
You could imagine a scenario where Doctor Who and The Avengers team up to save the universe from a time-traveling threat. Maybe there's a villain manipulating timelines, and both teams need to combine their skills and knowledge to stop them.
There can be all sorts of stories. Maybe adventures in different dimensions or romantic encounters. It really depends on the creativity of the fanfic writers.
One possibility is a tale where a Time Lord gets involved in the Game of Thrones' political intrigues and uses time travel to change the course of events. Or imagine if a White Walker was accidentally transported to the Doctor Who universe and caused chaos.
You could have a fanfic where they team up to face a common enemy in a new adventure.
You could have a story where Doctor Who travels to the magical world of Harry Potter and teams up with a powerful magician to solve a mysterious threat.