I am a board member of a Condo HOA there is a renter requesting a variance to the condo rules regarding pets living in rental units.

The answer will almost certainly be found within the text of your association governing documents.  Read them carefully.  If you do not understand the meaning of those provisions that seem pertinent to the question, consult a good real estate attorney.  The comfort of knowing exactly where you stand on the question is well worth a modest conference fee.   If the governing documents do not seem to address the question, the board should rule on the question.  In that event, it seems to me that three considerations should be at the forefront: (1.)   Restrictions on pets are probably designed in part to contain noise and other annoyances caused by pets.  What kind of pets are involved here and what is the likelihood they will annoy the neighbors?  (2.)   Restrictions on pets are possibly designed in part to help preserve the quality of the homes within the neighborhood.  Homes are sometimes damaged and defaced by pets thus detracting from the overall appearance of the subdivision.  The owner should be consulted.  (3.)   Bottom line -- there would seem to be no discomfort in denying a