Hi Suzanna,
More than likely, your daughter has an inflamed tear duct caused by the virus also responsible for the cold. This should resolve on its own as her immune system clears the virus over the next 7 - 10 days. Viruses CANNOT be treated with antibiotics, and using antibiotics when the infecting pathogen is a virus can contribute to development of antibiotic resistance in those organisms (bacteria) which do respond to them. Please Mamas, practice good antibiotic stewardship, or someday soon we won't be able to treat even the simplest of bacterial infections, and we'll be back to the 1920's, when people died of simple infections like strep throat.
You can treat this locally with steam and warm compresses, along with a mild decongestant (like phenylephrine) and lots of fluid to help thin the mucus that's accumulated in the duct so it can drain through the sinuses and nose, appropriately. You can use children's ibuprofen (motrin, advil) to help relieve your daughter's fever, pain, and inflammation. If, however, her fever suddenly climbs to over 103 deg. F and/or if the pain becomes severe, she can't tolerate normal light, or you see greenish pus oozing from her eye, these may be signs of a bacterial infection, which would need to be treated with antibiotics, and you should take your child to see a doctor. You don't need to go to an ER, however. An urgent care center would be more appropriate and, depending on your insurance, less expensive. BTW, most pediatricians have partners who are available or on call when they're out of the office. They rarely leave their practices unattended. Have you tried calling the doctor's answering service?