Power converting - up or down? Sorry if there is an obvious answer to this - I'm pretty new at Arduino and electronics. I've searched a bunch, but I think I'm too vague - and don't know what terms to look for.

I have a pretty simple project using a 3.3v Arduino Pro Mini (https://www.adafruit.com/products/2377) and a 12v button/LED (https://www.adafruit.com/products/1185).

I'd like to give the LED the full 12v so the light is a bright as possible. The system will be battery operated.

The question is: Do I need a 12v power supply or can I use a 3.7v power supply and upgrade it somehow? I know I can power the LED with 3.7v but then it's really dim. Is there a way to "upgrade" the 3.7v to 12v? I would assume this would draw more power and drain the battery faster, but I think that would be ok.

I'm hoping to get away with a 3.7v power supply because I want to use a lithium ion polymer battery (https://www.adafruit.com/products/328) - since it's small and rechargeable.

Seems like if I go with 12v source then I need a much bigger battery pack, and I'm hoping to keep the project small. Not sure if it would help to know, but I only need the device to run for 4+ hours between charges.

I'm guessing people will say to go with 12v since the Pro Mini can downgrade the 12v supply so there'd be less parts I have to deal with (I can already make this work). But, again, I was hoping to get away with a small rechargeable power supply. Or, are there good, small 12v batteries - I can't seem to find any on Adafruit or SparkFun.

Any recommendations? Thanks!
Since your project is to be battery powered, you want to make your system as efficient as is reasonable.  

So I am going to take an approach that ends up improving the efficiency of your project.  

The button you describe in your link states :  


  This button has a domed red plastic cover with a black retaining ring around it. There's an optional LED included  


Further, the LED included has a "built in resistor", so that the LED can operate from 12 volts. 

Since you would have to install the "optional LED",  why not purchase a Red LED that operates from lower voltage. You will still need a resistor but you won't need to have a voltage converter.  

Example  :  Their LED probably has a 2 volt LED (approximate) that operates at perhaps 20 mA).  This would require a 500 ohm resistor (which they incorporate into the LED).  The power loss (wasted) in the resistor is 0.2 watts (200 mW).  

Improved Example :  With LED of about 2 volts operating from 3.3 volt power source.  This would require a 68 ohm resistor.  The power loss in the resistor would be 0.027 watts (27 mW).  A substantial improvement in efficiency.  

The manufacturer of the button does not give specifications for the LED (the one with the resistor included).  You should connect their LED to a 12 volt power supply and measure the current drawn.  It is possible their LED draws more current than I have shown in the examples.  This would require a re-calculation of the resistor values.  

But in any case, the improvement in efficiency will still exist.