Welcome to the forums!
A few thoughts:
First, if the computers are physically broken (failed hard drive for example), using Linux instead of Windows will not magically repair the physical damage.
Second, you need to tell us the hardware specs (CPU, RAM, video card, etc.) to get any meaningful recommendations. If the hardware is modern (1ghz+ CPU, 1gb+ RAM, supported graphics card) then Mint is a very popular distribution for first time Linux users.
Third, you should find out which operating system the recipient of the computer actually wants. Most people prefer Windows over Linux and furthermore may require Microsoft applications such as Office for work/school. Therefore my recommendation is to troubleshoot the Windows install (a fresh reinstall often works wonders) unless you know for a fact they want Linux. If that is the case, then I would say, choose the Linux distribution that you personally are experienced with so you can provide the necessary support.
Good luck and please feel free to post back with additional details/questions!
Last edited by snowpine; 07-17-2012 at 07:34 PM.