I've tried searching for the correct driver but Linux Mint 10 does not include one for the HP Deskjet 3000 J310 printer.
I tried to install the correct driver on linux mint 10 last week and I have a HP Laserjet 1200. I think you don't have to install anything. Just look if you have the HPLIP package. On my desktop, the ubuntu version was installed automatically, but it works fine.
So you can just add the printer, without installing anything.