Electrical – Does it matter if the newly installed GFCI reads “open 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?

Check for a neutral to ground short somewhere on the load side of the GFCI and downstream outlets. Also, make sure there isn't a shared neutral with any other circuit and that you don't have a connection from the neutral load directly to the line side of the GFCI (in other words, the whites cannot all be pigtailed together, the line and load neutrals must be separated by the GFCI).