Linux Mint Maya is now installed and running on my new ASUS eee PC model 1015BX.
It came pre-installed with Windows 7 Starter, but I wanted a Linux only computer. The first installation try with Linux Mint Maya caused the computer to stop booting. UEFI was creating new challenges, and apparently UEFI has its own partition by default from the factory. The Linux installation wiped out the partitions on the hard disk completely.
I messed around with the program gparted from a live USB disk, and now the computer is able to install and boot with a msdos partition table. I now have 2 partitions (or is it 3?…):
/dev/sda1 ext4 297.36 GiB
/dev/sda2 extended 745.20 MiB
/dev/sda5 linux-swap 745.17 MiB