how to remove a tiny screw that won't spin?
First, you need the right sized screwdriver bit - this is already looking chewed up as though it's met not quite the right size. This is one area where getting the non-Chinese product may pay off in the long run (they not only work better, they also destroy less things you work on, so between not having to throw out and replace the tools and not having things you work on damaged, the more expensive tool may cost less overall.)

Second, you need plenty of force pushing down on the screwdriver so it won't pop up. A standard "jewelers screwdriver" has a rotating button on the top so you can push down while turning.



You'd need some adapters to get down to the correct bit size, but a hand impact driver is the tool of choice for truly stuck cross-head screws - preferably with the right bit - be it Phillips, Pozi-Drive, JIS or some other variant on the + screw head (they are different, in ways that matter) in the right size. You torque the body in the direction you are trying to turn to set it, then smack the back end with a hammer, which drives it in and turns it at the same time. It's rare to need to go that far on a laptop, though, and you could cause additional damage if the computer is not well supported.



Third option would be to accept that the screw is stuck and it's not your laptop, and suggest that the friend take it in to a service tech (sometimes discretion IS the better part of valor.)

Fourth, you could (with some damage to the black plastic part) cut a slot in the screw. Messy and complicated (can't let the grinding dust go everywhere, access is poor, etc.) but it gets you more leverage (wide screw-head, small + type driver - can use a wider straight-blade on it once slot is cut.) But look very hard at the third option before going there.