I weigh my babies and youngsters once a week - it's probably overkill but I like to see how they are doing weight-wise. I also chart their weights - but the less said about that the better as I don't want to seem like some nutter
My older animals I weigh every other week or once a month - just to keep an eye on their weights - my older ones all seem to lose rather than gain weight and I think that's often the way with older hamsters.
As Basia says Syrian weights vary widely - breeders breeding for "type" in show hamsters breed for a heavy "chunky" look which obviously weighs more than the skinnier pet-shop type hamsters. My lowest weight is 118g (Treacle who has had two strokes) and my highest is Caramel (190g and 10 month old female) with Mars (189g and a four month old male) close behind!
Females are usually larger and heavier than males, also as Basia says.