Watery eyes are quite common in the daily life and can be the result of many a cause. As far as I am concerned, dry eyes, clogged tear ducts, conjunctivitis, allergies and many other environmental irritants may all lead to watery eyes. You said your eyes water when you are sleeping and I am sure that you can check whether one of the above-mentioned reasons is the main source. By the way, over-the-counter remedies such as artificial tears and medicated eye drops can help release the watery eyes. If your eyes keep watering for a long period of time, it's the right time for you to see the ophthalmologist.
The eyes infection may cause your watery symptom when you sleep. You need to use the eyes drops to release the symptom. The eyes drops should be owned with anti-inflammation role. You should not use the eyes too much time every day in front of the computers. You could use the warm compress to make your eyes feel comfortable.