I'd have to see a picture to get a better idea what could be wrong with the eye. How are his teeth? Dental problems are quite common with hamsters because their incisors constantly grow. Overgrown and or misaligned incosors can curl and grow into the jaw, cheekbone, below the eye sockets, and pretty much anywhere in the mouth. This would explain possible eye issue, weight loss, and lethargy, since eating in this condition would be painful. Also check and make sure he does not have a wet bottom, which would mean he has a developed wettail, a stress related illness characterized by loose stool. Many hamsters develop wettail because of stress, and if caught soon it can be very treatable.
If his teeth are overgrown, curling, crooked, missing, or anything different than normal, take him to an exotic vet to have his teeth trimmed, and make sure he has enough to chew on and the correct diet.
I'm hoping this is the issue with your hamster, even without seeing it, since it is something that is fixable. Good Luck!