I'm still shopping around for a good basic laptop for my wife.
I've had good success just buying whatever is on sale at a computer store near me, and then wiping it and installing Linux.
The last time I did this, I bought a Lenovo IdeaPad S415 for something like $350, brand new. And to my horror, Linux installation failed on it; it includes both an AMD A6 and a discrete graphics adapter, and the two graphics systems fatally confused X11. There were workarounds but I never got around to trying one.
Almost a year later, I simply grabbed the latest Linux installer, and the install Just Worked. Someone had patched whatever the problem in X11 and everything worked: the touchscreen, the WiFi, the sound, the Ethernet jack, sleep on closing the lid, everything.
The moral of this story: most of the time, a Linux install will Just Work. But if you really need to be sure it will work, I suggest doing a Google search for the name of the laptop plus the name of a popular Linux distro. When I searched for "Lenovo IdeaPad S415 Ubuntu" I immediately found discussions, by other people who had the same problem I had, and the workarounds they figured out.
At the moment, my father is using that laptop as his main computer. He's using it for hours every day. We are getting our money's worth.
Not having to pay the Redmont tax is an even bigger deal on a cheap laptop.
I'm not sure, this may be changing, but historically low-end computers are choked with pre-installed software you don't want, and the computer maker collects money for installing this bloatware. Enough money that they make more than they spent on Windows... at least in some cases, you actually would have to pay more to get a Linux computer.
So unless you are really upset with Microsoft and don't want to give them a dime of your money, just buy whatever laptop is a good value, and wipe-and-install.