That switch has a single wire, so it’s hard to tell what its function really is. Perhaps it’s mains AC. Perhaps something else.
I suggest you open the case of the switch, and show a picture of the insides. You should also take a multimeter to the switch, and measure the voltage across the switch when it’s off, and if possible, the current through the switch when on. Just don’t touch any live wires with your body!
Let’s assume the switch is mains AC, and that it runs something less than 900 Watts (so it’s < 8 A for 120, or < 4 A for 240). Further, let’s assume it’s a simple SPST switch that just turns “on” and “off.”
In this case, you can bridge the switch using a relay – either a simple mechanical relay with appropriate current/voltage rating, or a solid state relay (“SSR”).
SSRs can be driven directly by an Arduino. For example, this guy, plus a 180 Ohm resistor, will work fine:
http://search.digikey.com/us/en/products/S108T01F/425-2395-5-ND/720455 (about $5)
(This is the 120 V / 8A version; it comes in higher voltage/currents too for about a dollar or two more – I’m using it myself and I love it)
An alternative is a mechanical relay, which then also needs a MOSFET or other buffer from the Arduino, because the relay coil draws too much current.
Now, wire a 180 ohm resistor from Ardy output pin to “+” side of the SSR, wire “-” side to ground, wire a 47 kOhm pulldown between the ardy pin and ground for good measure. When you drive the output pin “high,” then the relay will conduct AC across the AC-marked pins. Connect those AC pins across the switch on the inside of that black box, and you have what you need!