best (inexpensive home) tool to cut thin brass cleanly?

One word: Dremel.

The best way to do it, on either a scroll saw or a band saw, is to glue the brass between two layers of wood. Make sure that you have the right solvent to remove the wood later! The wood will give you a sturdier object to hold on to, and will prevent burrs from forming on the brass. The band saw blade won't walk if it is tensioned properly, is the right type of blade, and is properly calibrated.

LARGE tin snips?

Not sure how thick the plates are you're talking about - anything below 5mm (I don't speak inch) you should avoid band sawing or at least use a piece of flat masonite to absorb the vibration cutting will cause. Why 5mm, well standard sawing practice recommends a minimum of 2 saw teeth touching the surface of the job at all times or there'll be a tendency for the teeth to dig in vibrating the job and ripping out teeth (the finer the saw blade, the better your chances - won't lie I've cut .5 material when noone was looking but the blade doesn't love me for it). As for cutting straight - I don't think you have too much to worry about, thin material doesn't offer as much resistence so you can pretty much control the cut but I'd recommend using as wide a blade as possible with guides keeping the back of the blade (behind the teeth) straight so that it won't have the chance to wander off course. I won't go into the safety of it though - bandsaws favourite food... fingers! we have a small foot operated guillotine here that we cut brass plates around 0.5 mm on (could probably go up to 1mm but I'm sucking it out of my thumb here), when you start getting around the 2mm to 3mm thickness of material I'd say you're talking about a pretty heavy (probably motor run) guillotine - yes its only brass and not steel but I don't think that'll make it any easier to cut guillotine wise. Back to the band saw - if you're looking for a lubricant, try paraffin. Good luck

disk saw thin sheets - guillotine (like the 2 guys on the bottom said)

http://www.micro-tools.com/store/item_detail.aspx?itemCode=SAW-313-00
jewelers saw for cutting thin metal.

40 tooth or better tungsten carbide tipped saw blade, look for a thin kerf, for a 7 1/4 circular saw or bigger (( sliding compound miter type would be the best )) now I'm not 100 % sure how well it would cut brass but they sail right threw thick aluminum even large heat sinks, and if you use the guide or a fence your cuts should be straight ...

Tin snips?

. Clamp the brass between two sheets of hardwood or metal. Use a fine-tooth saw to cut all three layers at once.

An old style guillotine will cut thin sheet metal, the ones that look like a big heavy sword on a hinge... You could fold it and cut from inside the fold...

Experiment with a heavy duty paper cutter. The ones that have a big single blade that chop down. It might work with thin sheets of brass if you just want to cut a straight line. Maybe some sheet metal brakes or formers have a cutter built in.