what would cause brown black stains on the gums of a child of 3 years

Sometimes a child's mouth become colonized by a strain of bacteria that exhibits a color stain.  The teeth need to be cleaned and then purged of the colored bacteria.  Sometimes they are very persistent.  This may be a job for your dentist.  One way to change the oral bacteria population is the long term 4x/day use of a Xylitol chewing gum produced by Epic or Spry, or Emerald Forest. 

Check out these links:

http://soundentistry.squarespace.com/blogodontics/2009/10/20/xylitol-chewing-gums-reduces-cavities.html

and

http://soundentistry.squarespace.com/xylitol-stops-cavities

 Good Luck!--Dr.R.