Depends on your system.
If your machine is quite a bit older it may be difficult to run it as a media server.
A good OS for older hardware is Puppy Linux.
A media server in your home isn't going to be more than a file share for you alone so I wouldn't assume a lot of security is necessary.
You could really use a wide variety of Linux distros. All you need is to share files from computer to server and vice versa.
I like Debian personally.
ClearOS is good.
A lot of people like Ubuntu. (it was based on Debian)
There is CentOS also. Cent OS is basically RedHat without having to pay for support. After RedHat releases their code changes the CentOS team apply it to their OS, and keep building off of it.
If your desktop is Windows based you can always use Samba and Putty to help run a Linux server with it.