I can only tell you how to back up the background image.
On your brother's laptop
1) Open a terminal and type: sudo -i (it will prompt for your password)
2) Enter in grub's folder: cd /boot/grub
3) List all visible files: ls -l
4) Find any file with image extension *.jpg, *.png, *.gif or *.tga (probably *.jpg)
5) Copy the image to usb hard drive:
cp background_image.jpg /media/user_name/usb_hard_drive
On your laptop
6) Repeat step 1)
7) Copy the image back from the usb hard drive to your grub folder:
cp /media/user_name/usb_hard_drive/background_image.jpg /boot/grub
8) Now reboot your laptop and it should load your image automatically.
And here you have how to save the last grub choice BUT be AWARE if you don't know what are you doing you can make your grub INACCESSIBLE!!!
How to get grub2 to remember last choice?