The name probably comes from sailing vessels, where the fore-aft lines that hold the mast vertical are called "stays". If the bridge deck were the vessels deck, and the bridge tower the vessels mast, then the cables of the bridge would resemble the mast stays. Never mind the function is practically reversed! The bridge tower supports the deck with the cables - while the vessels lines hold the mast in place.
Like a suspension bridge, the cable-stayed bridge uses cables an towers to support the deck. But while they can be confused and might look similar with lots of spidery cables and tall towers, structurally they are quite different.
No bridges are currently listed in this category.