The biggest concern is how much ram and how fast a processor of the older computer. Most modern distros use about a gig of ram on startup and prefer a processor made in the last 20 years. If your computer has 500mb ram and a single core 1ghz pentium its gonna choke trying to run linux mint.
Instead certain Linux distributions are specifically tailored to work on extremely old and underpowered computers such as puppy Linux. These are modern distributions with updated kernels but are extremely minimalist in nature.