I don't have an HP mini, but have two eeePCs. It is easy to install Ubuntu or
many other distros to a SD card or usb stick. The trick is to write grub to the
SD card rather than the Hard drive and to control the boot process by using the
boot manager that comes with your PC.

In the Ubuntu installer choose to manuallypartition and make sure that your SD
card is inserted before you begin the installer. Choose it by clicking on the
ribbon graph or by clicking on the table. It will likely be called sdb or sdc,
depending on the number of drives, but you can go by the size which will be
close to the capacity of the card or stick. When you manually install use the
whole disk. Right click and choose Edit. Choose ext3 as your format type and the
mount point will be /. The rest is normal. On page 6 choose your user name and
password, but on Page 7, go to the bottom and click the Advanced button. There
choose to install grub to sdb or sdc, whatever the card is. Do NOT write to your
sda. Then commit it and wait for it to finish.

Afterwards when you boot you need to have the SD card inserted and in my HP
desktop and my eeePCs you press ESC to get the boot manager to kick in. There
choose the SD or usb device to boot. Grub will load and it should work. If not
you will have to manually edit grub.

Possible problems could be that you mistakenly write to sda. You should consider
entering the BIOS before you install and change it so that your hard drives are
not disabled. This is no guarantee of protection, but if you do it, don't forget
to enable them afterwards. Also if you have any sensitive data or bookmarks etc
you might want to back it up first to another usb stick. You can never be too
cautious. The worst case scenario is that you have to recover the original
installation to keep your warranty intact and that will mean loss of data.