Even if there is a graphical display, if you don't change anything, any operating system is unlikely to use much CPU for the GUI. The hungriest resource for rich GUIs is probably RAM, and perhaps battery power for the GPU if composited/accelerated.
I use Arch linux with LXDE for a similar requirement, on ARM computers and x86/64 computers. Arch takes a little bit of learning, but it allows you to completely configure the set of software running, so system resources are available for what you want to do. LXDE is not a composited/GPU-heavy desktop environment, so in that sense it's lightweight on system resources.
You can also use Ubuntu, or Mint, or Debian, in the server/headless configurations, and then separately install XOrg and LXDE to get a very similar setup. I think you can also do this on RedHat/Fedore and Gentoo and a bunch of other Linux distributions, but I'm less familiar with how those do package management.
That being said, if you want to run Windows 7, just switch the desktop settings to "basic" and you'll probably do fine there, too.