Chalazions in toddlers occur in two forms, the "internal" variety and the "external" variety. Your description is not totally clear, but it sounds like this bump is trying to drain on the skin side, which would make it an external chalazion. These do leave small skin scars when they drain which usually fade away to almost invisibility over time.
Warm wet washclothes used as compresses are germ factories, and ineffective as well since they don't stay warm long enough to melt the thickened secretions inside the chalazion. Since most chalazions are impacted and inflamed, but not infected, I typically do not prescribe antibiotics either orally or topically.
I think you should first call your child's pediatrician with an update. You might consider taking him to a pediatric eye-MD.
The #1 cause of chalazions in toddlers and children in my opinion, having cared for kids for 15 years, is excessive high fructose corn syrup in the diet.
To Dr. Sandra Brown,
My 3 year old daughter has a chalazion that we are planning to have surgically removed since it has been there for over 7 months. It is a difficult decision to make since it does not bother her and most of the time it is not too noticeable. At times it gets red and inflammed and then in a few days it flatens out. We have done 2 rounds of antibiotic drops which changed nothing and did compresses the first 3 months or so which changed nothing.
My daughter's diet is AWESOME. I have NEVER bought or prepared food with ingredients that contain high fructose corn syrup. I read ALL labels. We rarely eat out and when we do she is a fan of grilled meats and green beans. She drinks organic fruit juices diluted 75% with water.
Children's diets are a BIG issue that Pediatircians should address but your comment and OPINION of the NUMBER 1 cause is so outrageous and without any evidence. It makes me wonder if you are really a doctor. Folks have had chalazions long before high fructose corn syrup came around. As a doctor, it is important you mention what may be the cause but you really should keep such ill-advised conjecture to yourself. GUILT on the parents in this case is unwarranted...this chalazion is not type 2 diabetes after all.