One of the things you always want for a remote machine is, if not the ideal of LOM (Lights Out Management, see below) is a serial console or at least serial terminal.
Pins 8 and 10 on the GPIO header are the TX/RX pins for a 3.3 V TTL serial interface. The usual way of connecting it is to use cable with a 3.3 V serial to USB converter built in and plug the USB end into a nearby computer. The Adafruit
Using a Console Cable guide gives details of how to get up and running. Note that this actually sets the boot console to the serial port, so you'll see all your kernel and startup output there instead of on the monitor when you boot the system. If you simply want to have the serial port as an extra terminal while still having the keyboard/video as your system console, you can do that by enabling the "getty" service on it.
This blog post includes information on that, along with some notes on the changes to serial configuration with the Raspberry Pi 3.
If your computer is more than a couple of meters away from your Pi you'll need to run the serial link further, which you may be able to do at low bit rates (say, 9600 bps) simply by running three longer wires (TX, RX ground) from the Pi to the USB converter. At longer distances even this will be too error-prone, however, so you'll want to hack together a converter the Raspberry Pi end to convert the 3.3 V TTL serial to something like
RS-422 and a converter at the other end to bring it back to TTL for a USB serial interface. This can give you a range of a kilometer or more over four wires, which is likely to reach even from the deepest basement to the furthest attic.
Full Lights Out Management is a common thing on rack-mount PC servers: it adds a second (small) computer on to the motherboard that usually shares an Ethernet port with the main system but can be contacted independently and is powered up even when the main computer is down. It lets you power on and off the main computer, see the boot screens, configure the BIOS, and so on, all via a network connection, and it's not uncommon for the only actual server access to be installing it into a rack and plugging in the power and Ethernet with the OS install and everything else from that point on being done remotely.