My father had this POS laptop for years, and it finally crapped it's pants. I made a build on PcPartPicker for the budget of $400-500. I don't want to spend $500, but I will if it makes a significant performance boost.
Hm, would not get an A10 unless casually gaming. I would swap the A10 with an A8, Still a quad core. Or possibly just getting a regular CPU and relying on on-board graphics (Not sure about that though, I always put in a GPU)
As for the CPU cooler, I would go for a 212 plus or the 101(AMD) that is listed under the price.
Would use the ASRock FM2A75M-DGS instead of the ECS A75F2-A2, Just a personal preference, nothing wrong with the ECS A75F2-A2 as far as I can tell.
I would get a larger SSD, or a HDD along with the SSD. 64GB is not a lot, Win7 takes up 40ishGB. I use the Kingston HyperX 3K 120GB, It works alright.
Other than that, the rest is fine. It will be a good gift.
Get an athlon 750K CPU, it has a disabled graphics core which you won't be needing anyways. Even a celeron with Intel HD graphics will be ample for any browsing and videos. No need for a quadcore for your purposes. You don't need an aftermarket cooler for browsing the web. Also, gold certified PSU is overkill for this build. Get any 80+ power supply and you will be fine. Get 4 Gigs of ram in 2X2 instead of 8. You can find a cheaper case and it will run just fine. It's just a house for your components.