• 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?

Without being able to see the cables as they enter the cabinet; or the ability to touch or trace them, here is what I assume is going on.

Definitions:

Grounded (neutral) from the service

A typical single split phase service is made up of 3 wires. Two ungrounded (hot) conductors, and one grounded (neutral) conductor. The ungrounded (hot) conductors will connect to the main service panel through a disconnect (usually a large breaker), while the grounded (neutral) connects to the neutral lug. The neutral lug will be bonded (electrically connected) to the neutral bus bar, and all grounded (neutral) branch circuit conductors will terminate at the neutral bus.

Grounding Electrode Conductor

This conductor is used to connect the grounding electrode (ground rod, etc.), to the grounding bus in the panel. All equipment grounding conductors will be connected to this bus.

Bonding Jumper

The bonding jumper is used to bond (electrically connect), the un-energized metal parts of the panel to the grounding system.

Assumption:

Since it appears that (what I assume is) the grounding electrode conductor terminates at the neutral bus, I'm also assuming that this is the main service disconnect. This leads me to believe that the neutral and grounding buses are bonded (electrically connected). In which case, technically, grounded (neutral) branch circuit conductors can terminate at the grounding bus.

So you have two options:

1. Terminate the grounded (neutral) from the new circuit to the grounding bus.

2. Move the green wire that is terminated on the neutral bus, to the grounding bus. Then terminate the grounded (neutral) from the new circuit, to the freed up slot on the neutral bus.

Additional Information and Code Compliance:

Number of Conductors

Since this is a new circuit, it has to be installed to current code standards.

National Electrical Code 2011

ARTICLE 250 — GROUNDING AND BONDING

VI. Equipment Grounding and Equipment Grounding Conductors

250.140 Frames of Ranges and Clothes Dryers. Frames of electric ranges, wall-mounted ovens, counter-mounted cooking units, clothes dryers, and outlet or junction boxes that are part of the circuit for these appliances shall be connected to the equipment grounding conductor in the manner specified by 250.134 or 250.138.

Which in this case means installing a NEMA 14 receptacle for the dryer, and a proper grounding conductor.

You'll have to follow the dryer manufacturers installation instructions for upgrading to a 4 wire cord. For more information see this answer, and this answer.

Since you've said that you're already using 4 wire cable, you'll simply have to terminate the grounding conductor in the cable to the grounding bus in the service panel. Then connect the other end of the grounding conductor to the grounding terminal in the dryer receptacle.

Size of Conductors

You'll also want to be sure that you're using the proper size breaker and conductors. In the case of a dryer, you'll typically use a 30 ampere breaker and 10 AWG conductors (depending on the length of the run). However, you'll want to check the dryer manufacturers installation instructions to verify this.