fan?

You could. It would describe it clearly and accurately.

"Homemade hand-held fan" is okay, perfectly grammatical and understandable. However, calling a fan "hand-held" is sort of hitting the listener on the head, since that type of fan would have to be held in the hand. You'd more likely hear something like:

a homemade fan made out of a palm leaf

a homemade fan made of newpaper

a homemade fan

fanning himself with a

I've thought about this some more, and, at the risk of sounding like I'm nitpicking, I have to say the this whole thing - "a homemade hand-held fan" - is not right if he's just using a newspaper or leaf to fan himself. The reason is that the word "homemade" suggests a much more elaborate item, something that has been made to resemble a store-bought fan - the word "makeshift" is what you would use

I pictured a fan made by pleating a sheet of paper or by plaiting palm fronds.
If no effort went into the making of the fan, e.g if it were simply a newspaper folded in half, then makeshift would be appropriate. If creating it actually involved some effort I don't see why we shouldn't call it homemade. I don't think that home-made necessarily has to mean elaborate o