The movie is based on the book, The Exorcism of Anneliese Michel, and indeed, that was also the movies working title. I'd wager that they changed the name to Americanise the title in light of its primary audience. Anneliese Michel sounds decidedly European while Emily Rose is very American.
It's also possible that they didn't want to run into legal issues with the author of the book (which was published in 1981).
While the movie may be based on the true story, it doesn't neccessarily need to replicate it exactly. In the end they didn't just change the name of the protagonist, but the whole setting from Bavaria to the United States. This was a natural consequence of the movie being produced in Hollywood and for American audiences. It would be pretty weird and unusual for a Hollywood movie to be set in Bavaria. So given that the setting was changed in such a significant way, why keep the original name of the protagonist, which wouldn't fit to the rest of the setting?