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The distro isn't the thing that matters; how you configure it is. I used to use DWM (the window manager from Suckless Tools), Surf (the web browser), st (the terminal emulator), and dmenu (the application launcher).
They're all very well suited for not using a mouse, although the web browser is going to be hardest part. Surf has a patch that allows you to use keyboard shortcuts to control it, but it's still kinda janky. DWM is perfect for keyboard-only, though.
