I'd go with this as a first step:
ls /etc/*release
Gentoo, RedHat, Arch & SuSE have a file called e.g. /etc/gentoo-release. Seems to be popular, check this site about release-files.
Debian & Ubuntu should have a /etc/lsb-release which contains release info also, and will show up with the previous command.
Another quick one is uname -rv. If the kernel installed is the stock distro kernel, you'll usually find the name in there.