I have that same unit, and it works fine for me with Mint 5 and 6. I did have a problem with scanning at first, but I've forgotten the details. What I know works for me now is this: install XSane, then control the scanning function from within that (I don't think scanning works if you try to scan using the buttons on the HP itself--though copying does--but I'm not in front of my home unit to check). Basically, it's as easy as making sure the printer/copier/scanner was on, opening XSane, setting the type of file and save location, and clicking. When you see the image come up for review after the scan has been done, don't forget to click Save.
XSane should be available from within mintInstall, or if it's not, through the Synaptic Package Manager. You don't need the Terminal to install it, or to work it, and you don't need to compile anything.
If the 5280 still doesn't scan after you've got XSane up and running, I'm afraid you'll need someone more expert than I. Fortunately, there's no shortage of such people on this forum.
Olivia 15 Cinnamon 64, dual-boot Win 7 Pro 64 SP1, Clevo W150ER 15.6" laptop, i7 3630M, 16 GB RAM, Samsung SSD 840 Pro 256 GB/HGST 1 TB 7200 RPM, Nvidia GT 650M 1 GB, USB 3.0 HGST 1 TB 7200 RPM, HP Photosmart C5280