Why does my baby have gooey eyes?

Could be infection . "gooey" eyes could be a sign of infection (conjunctivitis) or it could represent a blocked tear duct. If tears can't drain, it can look like a discharge from the eye. If a tear duct is blocked, you might see a small swelling under the eye, close to the nose. Although your doctor should confirm this diagnosis, some gentle lacrimal massage (under the lower eyelid) may help open up the tear duct.
Blocked tear ducts. It is possible your baby has blocked tear ducts. To help these clear you can massage between the eye and nose on each side. This can help open up the tear duct and allow the tears to drain more effectively. The goopiness will go away in a few weeks or months. In the meantime use warm washcloths to clean. If the white part of the eye gets red or the discharge increases see your pediatrician.