What type of capacitor is most stable for use in 555 timer circuit?
As DKNGuyen said, if you want a stable capacitor, use C0G/NP0 ceramic.

However, if your actual goal is to have a stable frequency rather than discuss capacitors, then your original idea of using a 555 timer is not the way to go, as this chip will have worse drift than the C0G capacitor.

A much better option would be to use a quartz oscillator, for example a 2.4576 MHz 
oscillator which will cost about €1 in 50ppm/°C stability, then divide by 4096 using a 74HC4040 ripple counter. You could use a 74HC4060 too, with a crystal instead of an oscillator. Also the quartz oscillator is pre-calibrated and you don't need to adjust it.

EDIT I misplaced a decimal point...

I mean, a 1.2288 MHz oscillator followed by 74HC4040 to divide by 4096, resulting in 300Hz. Then a 74HC390 or similar which can divide by 5 then 2 then 2 which gives 60Hz, 30Hz, 15Hz.

Or a 192kHz oscillator then the same two 74HC chips, but this oscillator is only available on digikey in tiny MEMS flip chip so maybe not the best option. Anyway you get the idea, pick a frequency and a convenient division ratio with 74HC chips...

The BOM cost for both solutions will be less than a precision capacitor and a potentiometer, and that doesn't count the salary of the intern who gets to tweak the frequency just right...

If you need better stability than what a cheap XO will provide, you can use a TCXO. That will cost a bit more (a few €) and it will be available in less convenient frequencies. So you can use, for example, a 12.288MHz TCXO and divide by 60000, you'll get a few ppm stability over temperature.

Note a cheap microcontroller makes a nice programmable divider if you need one.

If you already have a microcontroller in your project, why not use that?