A hamster runs further than you think
Wild-type hamster behaviour does not shrink just because the cage does. In a single night a pet hamster will run the equivalent of many miles on his wheel, in bursts, at speed, mostly in the dark. That mileage is why wheel design is a safety question rather than a shopping preference: a small flaw in the running surface or the wheel's size is not something he meets once, it is something his feet and spine meet thousands of times a week. The two designs most often sold with starter cages — barred wheels and undersized wheels — each fail in a different, predictable way.
What barred and mesh surfaces do to feet
A barred wheel is a ladder bent into a circle, and a hamster's foot is small enough to pass between the rungs. At running speed a slipped foot can trap and twist a leg, wrench a toe, or in the worst case fracture a limb as the wheel's momentum carries his body forward over the caught foot. Even without one dramatic accident, the rungs hammer the soles night after night. Hamsters develop sore, inflamed feet from running on bars and mesh, a painful condition that can leave them reluctant to exercise at all, and the open design also lets a cheek pouch or a dangling piece of bedding catch where a solid surface would let it slide.
The same objection applies to mesh. If toes fit through the surface, the surface is a trap moving at speed.
What a too-small wheel does to the spine
Watch a hamster on an undersized wheel from the side and the problem is visible in one glance: his back curves upward into an arch, head lifted, because the wheel's diameter forces him onto a curve tighter than his body wants. Running mile after mile in that posture loads the spine unnaturally, and over time it can contribute to back pain and to a hamster who quietly does less of the exercise his health depends on. Starter-cage wheels are frequently too small even for dwarf hamsters, and almost always too small for a Syrian, who needs a notably larger diameter to run with a level back and his head in a natural position.
Size the replacement by watching the run itself: back flat, body straight, no upward arch. If in doubt between two solid wheels, the larger one is the safer judgement for the animal in front of you.
What to choose, and where to put it
Choose a solid-surface wheel with a closed running track, a smooth, quiet bearing, and a stable stand or a firm cage mount so it does not rock as he mounts it at speed. Place it on the firm, shallow-bedding end of the enclosure or on a low platform, and check nightly at first that bedding kicked against it has not jammed the spin, because a stuck wheel discovered after a week is a week of exercise lost to an animal who organises his night around it.
A hamster who ignored a barred wheel will often run enthusiastically the first night a proper wheel arrives, which tells you the old one was not uninteresting — it was unusable.
For a second angle on diameter and fit before you buy, put a follow-up question to KinsLe's Kinny; he answers using the KinsLe Answers library.