Electrical – Tester showing ground, but I am not seeing a ground

• Check for unused grounding wire, and connect if found (great option, but I suspect we will not find such wiring based on the age of the building)

This is the only option that will actually provide equipment grounding, assuming you find an equipment grounding conductor in the box.

• Replace receptacle with ungrounded GFCI outlet

While this will provide ground-fault protection; which is good, it will not provide equipment grounding. In fact, when you do this you have to add a label that reads "No Equipment Ground" on each receptacle protected by the GFCI.

• Connect a ground wire from the outlet plug to it's box (presuming we discover the box is metal not plastic) and use the box for ground

This only works if the metal box is connected to an equipment grounding conductor, otherwise it offers no grounding at all.

• Ignore the lack of proper ground and cross our fingers none of our electronics get damaged

This is probably the most common solution. Though most people aren't ignoring the lack of grounding, they simply don't know they need it.

• Run an extension cord from a properly grounded outlet in a non-original part of the house to where the electronics will be?

This is a temporary solution, but you'd have to protect the cords from damage and use the proper size cord.

• Something else?

There's really not an easy solution here, the only way to do this is to run an equipment grounding conductor.

Other useful questions:

• Is it OK to borrow a ground wire from a different circuit?

• Can I run a ground wire of a single circuit to the panel outside the house?

• How can I determine if my outlets can easily be grounded?

Assuming you are in north america: In a home wired up with armored cable, otherwise known as BX or AC cable, the armor is the grounding conductor. It is always terminated to a metal box with a cable clamp that secures the cable and completes the connection between the box and armor. The metal mounting bracket(called the yolk), which is the frame of an outlet, is not only mechanical but also serves as a ground conductor for the outlets ground prong(s), the grounding terminal screw and the cover screw. The ground connection is made in one of two ways: via the ground screw or the mounting screws which secure the yolk to the box. The yolk method is acceptable in armored cable systems.

Newer MC or metal clad cable looks exactly like armored cable but has its own grounding conductor. In that wiring system, the armor is not listed for grounding and the wiring methods are similar to that of terminating non metallic cable (aka romex). Meaning, the ground wire must be connected to the metal box, the outlet, and any other grounding conductor.