You still could have called her Lilly. Start by using both names when talking to her and then see if she likes the new one.<quoted text>
Do you have a pet?
If so, what did you name it?
How did you come to the conclusion of the name?
Of I had my choice, I would have named Beany, Lilly!
Too late for that now I think but Beany is good. You can use it in public without embarrassing either of you. And how is Beany? Well, I hope.