What causes a young baby's eyes to be teary and have some mucus, when he isn't looking ill?

Usually tear ducts: There can be many causes for tearing and mucus in a baby's eyes. If those are the only symptoms/signs then a common cause are tear ducts that have not fully opened to allow drainage. Usually we recommend massage of the tear ducts and observation. If a conjunctivitis develops then the patient may require antibiotic drops or ointment. Your pediatrician should be able to examine and advise you.