I used to have a tiny cordless fan that had (non-rechargeable) batteries. My grandmother discovered them and thought they were great, so as they weren't expensive she distributed them around the whole family. It was OK but I didn't like using up non-rechargeable batteries. They didn't last all that long.
I guess with a rechargeable one, a lot depends how long the battery lasts before it needs charging, and how long it takes to charge. A lot would depend on the user's particular needs - are there convenient places to clip it onto, how good is the clip, that kind of thing.
In my case I don't leave the house very often so I'm content to use a small plug-in fan next to my favourite chair. I'd like one by my bed, so I should probably get another one for that. Meanwhile I use a small space-heater set to just blow without heating. It's rather weak but it's better than nothing when the air feels stuffy. If I got a rechargeable fan for that, the battery would need to last about 10 hours, or it would conk out in the night after I'd gone to sleep. Though maybe that wouldn't matter as long as I didn't wake up till morning.