radioshack
Model: 275-233
Catalog #: 275-233
(didn't think the link to the search would show up right)
this is just an example of one. the current that they can handle may be the biggest drawback and yes they are supposed to be mounted in a PCB but so is the transistor that is being used. I think that current through the device may also be making it hot.
transistors are used for switching but the reason they use them for switching is usually for #1 no moving parts but #2 their switching speed is much faster. the second one goes along with the first but it doesn't have much to do with the size. a solid state relay is just a transistor (be it a big one) in a package that fits in a relay socket. it does have all the resistors and other good stuff built into a neat little package so you don't have to do anything but plug and go. they last longer because of the less moving parts which makes them very valuable in industry.
also, those boards that you can find the transistors on, look at them more closeley and you may find relays on them too. most are square and long (the small ones) maybe a qtr to a half an inch square by about 1 inch long. two pins on one side (the coil) and 2 or 3 pins on the other side (the actual contacts) they are used more than people realize in electronics. not necessarily computers but definitely for controllers. old PLCs used them for their outputs. they were very low current though. that may be a problem with them though in this application.
*edit* the way I see this is a low-tech problem with a high-tech answer. that is the reinventing the wheel thing. you want a switch to flip off so that the lights won't come on when the car is off. a relay is simply a switch. flip off the switch to the coil and the contacts open. low-tech but effective. nothing to figure out (other than current capcity you would need) if you use a 20 amp one, you have nothing to worry about but you can figure that out and use an appropriately sized one if you care that much. I guess also remember to use one with a 12VDC coil but that goes without saying.
*edit again*
http://relays.tycoelectronics.com/datasheets/OMR.pdf
here is one from mouser. this pdf shows a lot of things. it shows physical size but it also shows maximum switching rate. these things get really small but also the current handling is small as well. a lot of good data on this data sheet. I wouldn't use this relay but you see what I am saying about relays being small.
sorry for the book ^^^