Michael,
I think we need a little more information how your girlfriend uses her computer. Just surfing the net and email - then maybe a Chromebook (I have never used one, but they are inexpensive and seem to be focused on web browsing and email.). Does she need to do office work - documents, spreadsheets, power point, etc? Then something with more processing speed and memory is needed. I am going to spitball a little here, but maybe a Core i5 and a couple of gigs of memory? I could be way off as I always buy more CPU and RAM than I think I need as I always end up needing it! How portable does it have to be? Maybe she really wants a tablet and a small usb keyboard? Play with the latest at Best Buy, then shop online for better deals.
Since you want Linux, check out Craig's List for some older Windows machines. Maybe a Dell Core i5 or i7. Figure into your total cost adding more RAM and a larger hard drive (to replace the aging drive that comes with the machine) that you buy online. I have had good luck installing Linux on older Dell laptops. However, caveat emptor - do your homework on any laptop you want to make sure there aren't any hidden issues with installing Linux. Check the Linux laptop database, look at the issues people ran into wiping Windows and installing Linux and be very sure you are comfortable with the installation procedures. There are lots of recipes for installing Linux on older laptops in the net. Or, time the purchase with an InstallFest and get the experts to help you! ;) But still make sure there are drivers for all the hardware before you buy! Or your girlfriend will be disappointed when her sound or bluetooth or network does not work....and never never disappoint your girlfriend! ;)
Finally, really spoil her with a new laptop from System 76 - Ubuntu 16.04 fully installed, great support, great hardware based on my experience of buying on laptop from them! (I am not affiliated with System 76 in any way).
As far as passwords go, I suggest investing in a good password manager like Lastpass or KeePass2. KeePass2 stores the passwords locally, but you can easily move the data file to a new computer - works with Windows and Linux. Lastpass stores the passwords in the cloud, so new computers are not an issue. There are plugins for Firefox, chrome, etc. All you need to remember is one big password to unlock your password vault, and both programs will automatically fill in user name password fields for web sites, so you can update all your passwords to 64 character mosnters and never have to remember them. I use both of these on Linux on a daily basis and like them.
Good luck!
Mark