A relay would wear out pretty fast in this application. Assuming the thing is DC powered, a largeish power transistor (probably a darlington) is a good way to drive it. Since the LED-Wiz has sink-only outputs, you have to drive it high-side using a PNP type (a P channel MOSFET would also work). This should be reliable and pretty easy to assemble (it's 2 resistors plus the transistor). Wire it up as in the below image.
Q1 is a TIP107; you can get that at Digi-Key. Other PNP darlingtons can also be used and may be available at Radio Shack or Fry's. A non darlington PNP can be used but will require adjusting R1 and R2 and may end up requiring the LED-Wiz or optoisolator to sink too much current. A heatsink is recommended on the transistor.
R1 should be about 500 ohms.
R2 should be about 5-10kohm (make it about 1k when using a the MOSFET identified below) and can probably be omitted as the TIP107 has it built in (but you do need it when using a MOSFET, and it doesn't hurt in any case).
A P-channel MOSFET might not get as hot and can be substituted for Q1 with the changes identified above. A STP12PF06 looks like a decent choice. The terminals maps as Gate=Base, Source=Emitter, Drain=Collector. The darlington PNP may be a bit more tolerant of current surges and won't be static sensitive like a MOSFET would, though. Take your pick or try both.
This circuit assumes the LED-Wiz can handle a 30V collector voltage. I *think* it can, but the website isn't clear. Ask Randy. If it can't the circuit gets a little more complicated.
It also assumes that you're OK with tying your PC's ground to your solenoid 30V power supply's ground. This should generally be OK if you're powering nothing but the solenoid from that supply. Tie the two grounds together as shown.
Both problems can be solved with the addition of an optoisolator, forming the second circuit.
The +5V in this circuit is from the LED-Wiz.
R3 is about 1k.
U1 is an LTV-816 (in stock at Digi-Key).
There are other ways to build this if you can't get the parts I've identified, but the above should work. It'll drive just about any high current (up to ~8A) DC load. I don't know what the Q-Bert knocker draws, but I hope it's not more than that! 8A @ 30V is 240W!
Even though the TIP107 and STP12PF06 have one built in, I'd still recommend that you install the freewheeling diode across the knocker as per any inductive load to protect the transistor. 1N4002 or similar should be sufficient.