Creating animal models
If you look at the definition of a model organism as this can be found on the Nature Homepage, this clarifies quite a lot:


  An organism suitable for studying a specific trait, disease, or
  phenomenon, due to its short generation time, characterized genome, or
  similarity to humans; examples are a fly, fish, rodent or pig, whose
  biology is well known and accessible for laboratory studies.


So you want an organism, that:


can be bred in large numbers
has a sufficiently short generation time, so you can analyze several generations of it
is well characterized
sufficiently close to the organism you want to learn more about (mostly: Humans), so the findings can be transfered
can be mutated and has your gene/protein/etc. of interest
has been sequenced completely


Besides this general considerations, practical ones also play a role. If you don't have a fish facility on hand, you will most likely not start using zebrafish as your animal model.

There are certainly more requirements, see one of the references.

References:


Requirements and selection of an animal model.
Selecting appropriate animal models and strains: Making the best use
of research, information and outreach
Basic Principles in Selecting Animal Species for Research Projects