You can use almost any linux distribution as a router; You'd want to install
dhcpd, the DHCP server, and
dnsmasq, a DNS forwarder. Beyond that, you'd simply configure the system firewall to pass traffic from your LAN interface to the WAN interface (Anyone that's messed with
iptables is laughing at this statement. It can quickly get fairly complex). If you have a wireless card, you can also bridge that ethernet device with your LAN device to make the behave as one.
Depending on how manual you want this process to be, you might look around for distributions configured specifically to behave as routers. They will often include web-based configuration tools, to make management similar to conventional embedded consumer routers. Linux Mint may not be the best option for this problem, and if you can use something else, take a look at the following:
DD-WRT is a very powerful linux-based routing distribution, and can run on many consumer routers or on x86-based systems (read, most PCs). If you can use any distribution, I would recommend trying this package out.
pfSense is another very powerful router package, based instead on NetBSD. This is what I run on my router because there are a few necessary features which I needed for my setup (multi-homing, to be specific).