I've got a COM header on the motherboard. What interesting things can I do with it?
In the first instance, simply bring it out to a 9-pin connector on a backplate, if you think you might need it.
Typically, I use a com port to gain console access to devices like Cisco wireless access points etc.
Also hobby electronics projects etc.
Or an old US Robotics 14.4 Modem for teh lolz.
What connection did the old BBC turtle use?
Unless you have devices which need console terminal access, then you won't need it.
As a console application, I use PuTTY, but the real hardcode would just use 'cu' on their Linux system.