I'm not sure what type of solvent to use to clean a derailleur.
Clean the derailleur pulleys by holding the blade of your screwdriver against the sides of the pulleys and turning the crank. You can do this with the wheel in the dropouts, or with the wheel removed.
Brush the dirt off of your derailleur using a combination of brushes with bristles of varying length and stiffness, like the ones in the Finish Line Easy-Pro Brush Set.
Degrease the derailleur using Finish Line Speed Clean, a powerful degreaser that dries almost instantly. You won’t have to wait before applying lube, or worry that the lube will be broken down by excess degreaser that didn’t dry.
This is also a good time to go over the pulleys again. First spray down the jockey pulley with Speed Clean while turning the crank, then take your rag, hold it between your thumb and forefinger, and pinch the pulley while continuing to turn the crank. Any dirt that didn’t come off on the screwdriver blade will come off on the rag. Repeat with the bottom pulley. This is most effective if the pulley is still moist with degreaser when you run it through the rag. Because Speed Clean dries so quickly, ensure this by spraying down only one pulley at a time, so that the bottom pulley doesn’t dry before you’re done with the jockey.
Lubricate the derailleur by first spraying it down with Finish Line Dry Teflon Lube, then injecting Extreme Fluoro, a fully fluorinated grease, into all points of friction.
What are you trying to clean off it? If you're thinking solvents I assume it's chain grease rather than mud.
I prefer to use a water-based degreaser like simple green and an old toothbrush. If it's particularly bad I'll take it off the bike and dismantle it. This thread on BicycleTutor has much more detail on how you go about it.
A quicker and less effective approach is to squirt some penetrating oil like WD40 on to loosen the grease then wipe it off. That will get you running again in a hurry but it's not good for the bike.
Traditionally people have used organic solvents like kerosene but they're toxic and degreaser is just as effective. Petrol is the worst, it contains contact poisons as well as having poisonous vapour.
The best thing I've used is called "De-Solv-it", you can pick it up at most hardware stores. It's almost magical how well it works, just spray it on and let it sit for a minute or two. You may have to brush the heavy stuff but the grease just melts away. It also claims to be biodegradable. Here it is on Amazon: http://www.amazon.com/Desolv-Citrus-Solution-12-oz/dp/B000BQWVTQ.
My approach is to first scrape the worst of the gunk off. Then lean the bike up against something outdoors, where the floor won't be damaged by oil stains, slide some newspaper (one or two "sections" of a full-sized newspaper will work) up behind the derailer and let it lay against the wheel, to protect the wheel (both brake surface and tire) from solvent.
Spray the derailer thoroughly with WD-40 (this is about the only use I have for the stuff), simultaneously scrubbing with an old toothbrush.
Wipe dry, allow the WD-40 to evaporate a bit, then lubricate. I generally use whatever chain oil I'm putting on the chain, though there are probably better choices. Over-lubricate a bit, then wipe off the excess after giving it a chance to soak in a few minutes.
One could remove the idler wheels, clean them separately, and then force grease into their bearings, but that's a lot of work, and I'm not convinced it's worth it.
I refurbish a lot of older bikes to re-sell, and if the DR is really nasty I remove it first. Then, I soak it down with WD-40 to loosen anything that's stuck and start working it by hand; making sure all the joints move freely. Then... It's the Dremel (wire wheel) and the toothbrush till I get all the crud off and the stains, surface rust, and other crud off. Finally, squirt down with regular old detergent and give it a thorough wash. Use very hot water to rinse so it'll dry rapidly. Finally, lube all the pivots. If the idlers are sticky or unduly nasty, I'll disassemble the cage and idlers and clean them up separately, greasing the axles before assembling. Lot of work, messy. But the result is a like-new DR.
Adding to all the above: don't mind the 'solvent'. concentrate on the brush.
The solvent will not do magic. Trust me, simple green is the SAME as regular soap bar. Get an old tooth brush, scrub a bar of soap, scrub what you want to clean. rinse. repeat.
If you can't get the old toothbrush there, you will have to disassemble. or use a rag. there's no way something not-extremely-toxic will do the work.
An old shallow cooking tray full of petrol. Disassemble the derailleur, put all the parts in the petrol as you go. Pick them up one by one and scrub with an old toothbrush until clean, put each clean part on a paper towel to drain/evaporate. Then pick up each part and rub it with a dry paper towel before putting it down on a clean surface. Finally reassemble with correct lubrication - which is different for different areas of different parts. You probably want two or three basic lubricants like chain oil, light machine oil and a lightweight grease.
Don't smoke while you do this, or use litres of petrol at once. Do it outdoors and don't be leaning right over the tray breathing fumes. I find ordinary petrol a lot easier on the hands than any of the citrus-smelling solvents which do tend to strip all the oils out of your hands. A super-aggressive solvent isn't necessary.
The key is disassembly! You cannot get these small complicated units really clean without it. Squirting stuff into friction points, soaking or whatever else does not always flush grit, old grease or grime out it can just drive it further in. Not to mention your increased understanding of how the thing works when it comes time to adjust, assess wear or evaluate a new one.