Atom is much more powerful than ARM processors. So, if anything HP Mini will be better at video playback
In what? Atom isn't not that much more powerful than ARM, it's faster yes but not to the extent of saying "much more". ARM has been catching up lately to match or exceed Atoms, the Cortex A9 is the latest technology that has shown to be pretty damn good, enough to beat Atoms with even lower clock speed. Dual core A9 at 500mhz can match up pretty well against 1.6Ghz Atom chip, enough to beat Atom without a graphic card, source
here.
If he's watching flash videos, HP will struggle so, playback isn't a bonus either here, it's a plus for having flash capability.
Now for the OP, I suggest going with the CLV laptops like Inspirion 11z, Vostro 13. Hell you can even get Dell Studio 15" laptop for under $800, trust me, if its for class and writing papers, I suggest getting a bigger screen. 15-16" laptops can be found cheaper than 800$, good ones too. You should look at slickdeals, logicbuy to find Dell's coupons to get good deals.
What you might have to consider is the battery life, do you want something that last all days in school? You might have to look at Asus netbook/notebooks, they are usually pretty good with battery life.