felic:
I found it to be extremely hard to find a step-up converter that takes 1-5V and converts that to a fixed 12V.
I remember using a step up converter that had a pot and I adjusted it to output 12V while the input was 5V, but as soon as the input voltage dropped, the output voltage dropped, too.
Ok, my input may not be as educated as others in here as I am (perhaps like you) still amateur in electric engineering world, but I recently did large study of voltage regulators that fit into my projects and did learn alot and started with exact questions you asking. I can probably add couple points too. I am not sure if you a looking to find completely assembled regulator or building one yourself, I assume you are looking to build one yourself.
I think other important thing is efficiency, different regulators have different efficiency curves, some can be very efficient at higher current and be terrible at lower currents, others vise-versa, you need to keep an eye on that.
Another thing, if you trying to build voltage regulator on a breadboard, if it is a switching regulator then chances are it won't perform well or not fire up at all, use good short wires, perhaps protoboard will work better.
There are also seems to be two sub-typed of chips: 1 - regulator, 2 - controller, speaking in "plane english" the difference is in the number of components you need to use controller have all switching elements (e.g.: diodes, transistors, mosfets) outside, regulator normally has that all built in.
If you building voltage regulator from scratch for your project, I think choosing the right one is simple do this:
go to digikey.com (or other supplier)
go to voltage regulators (or voltage controllers) department, narrow down large list by input and output voltages, then by MAX output current, then by package and price (as some of them are quite pricey, up to $7/chip). You will probably end up with list of few regulators that fit your need, then one by one study their datasheet, see how many external components required and which circuit is easier to build. choose couple , build prototypes , test them and in the end you will end up with one that is perfect for what you need.
PS: I think it will be very hard to find decent 12V regulator that can take input voltage as low as 1v, but who knows.
for example, these may work: