I love that she comes out to sit on your son's hand with some encouragement, such an adorable little sweetie. Not sure about this, but she seems like she could be having an allergic reaction.
1. Could her bedding be too dusty?
2. Are the paper edges too sharp and could they possibly have poked her in the eye?
3. What are her toys and hides made of? If the hide is made from inappropriate wood, it is possible that she has been sleeping/ spending long hours in it surrounded by toxic fumes.
4. I found a link to the cage you mentioned:
https://www.amazon.co.uk/Little-Frie.../dp/B005CAYZ4A
Is this the one? If so, I strongly discourage setting up the top two floors, because it is too high up and they look very dangerous to me. She could fall off them and hurt herself.
5. The cage size is 78cm x 48cm, so it's a bit on the small side for a Syrian. Maybe you can consider upgrading her cage? With hamsters, it's the floor space that counts, not the height. Heights can be dangerous, as even without the two floors, she might start climbing the bars upwards and fall off.
6. What have you been feeding her? Someone suggested to me before that my hamster might have had a food allergy, so that's good to consider too.
I'd recommend you take her to an exotic vet if you are able too, the eye doesn't look good to me, but I'm just a beginner too so I can't really tell. Hopefully others can comment further.