Electrical – New dryer, old outlet

If you're talking about an E26 lampholder (the medium screw-in base widely used in the US), there are only two electrical contacts: live is at the back of the lampholder and neutral is the screw thread. While the light fixture itself should be grounded, it's not available via an adapter.

In an outdoor situation, ground protection is even more important than indoors; if your exterior circuit also includes exterior receptacles, it should already be covered by a GFCI. As I said in an another answer, that breaks the circuit if there's a ground fault, but without a ground connection, any current is going through you to ground, instead of through the ground lead. So no, it's not safe to use a two-prong to three-prong adapter and leave the ground on your lights disconnected.

I have seen exterior light fixtures with a grounded electrical outlet built into them; if you were able to change the light fixture, that would be the safest way to go.

• 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?