After some dissassembling, I fall on this connector.
It connects the motherboard to the screen.
Here is the part attached to the screen:
Here is the part attached to the computer:
Apparently the controller is a chip called MagicGraph 128XD from NeoMagic.
Even if it seems difficult to connect i to an Arduino (while we didn't speak about building some shield to make ineterfacable and overcome issues Arduino can't handle alone), does it look familiar to any of you ?