But, Can It Run Lichess?

My current system;
Mainboard: Socket AM2
CPU: Athlon II 240e (slow old dual core cpu)
RAM: 2 GB DDR2
GPU: ATI RADEON HD 5450
OS: Ubuntu 14.04
rest: does not matter

YES (cpu usage: 20-30%)

Searching for the lowest budget system that can run Lichess...

Your mind

? www.dict.cc/?s=Your+mind

I believe Lichess is a pretty light site, despite its many rich features. It is however a good idea to have a fast CPU with many cores if you want a strong computer analysis as Stockfish supports up to 256 cores (max for a x64 system such as Windows 10) and IPC gets better with each generation. I currently have an old FX-8350 and am planning on upgrading when the Ryzen 2 CPUs come out.