If having a fun name for your mixed breed is critical, may I recommend DNA testing?
http://whatsmydog.com/
We had our mutt tested and she came out as a mix of:
-English Setter
-Boston Terrier
-Tibetan Spaniel
-Unknown Terrier (likely Jack Russell or Pit Bull)
We call her a Tibostunk Terrietter
Now, as to whether she's healthier for being a mixed breed - who knows - she has terrible skin problems, but even at her age still acts like a puppy.
In fact, my parents found her on the side of the road on 2/13/99 age 6 weeks, so we're about to celebrate 12 years of love and friendship together.
I wouldn't take a dozen Morkie-Tzus, Boxo Apsos, or Cock-Teses to replace her.