I answered this in a PM, but for anyone else who has the same question, here is my advice:

There are a few ways you can go about it. One way would be to transcribe your name based on how it's pronounced. In this case, that might be "Fiibi" or "Fibii". 

Alternatively, you could come up with a dragon name that is similar to your own or has a meaning you like. "Phoebe" comes from Greek, meaning "bright/pure," so a dragon name like "Kunhindaam" ("light hope serve") could fit.

See the dragon name generator for more ideas on dragon names.