My 9 month old has red spots on and around her eyelids. The eyes also look a bit puffy at times. What could this be?

Irration of glands. It sounds like she has either dacrocystitis or blepharitis- both caused by irriation and inflammations of glands in the eyelids or tear ducts. Certainly start by using warm compresses to clean any drainage and use baby shampoo to clean the lashes and eyelids. Check with your pediatrician if things don't improve.
Eye rash. Are they small redish purple dots or big spots, it could be any number of things from petechia to allergy, i recommend seeing her pediatrician.