Anything will support that system as all it is is a serial connection with the HC-06 bluetooth module and a few GPIO to turn on relays.
I've been out of this for a couple years, so I'm not certain if there is anything better right now, but:
STM32 or STM32F4 discovery will work. Cheap development board which includes the programmer.
LPCXpresso. Never worked with these development board, though I have used plenty of LPC chips. I think the LPCXpresso IDE has some kind of simplified language. Anyway, it also comes with an on board programmer and there are different versions.
I was going to include a Freescale one, but it looks like Freescale and NXP merged a couple months ago.
EDIT: Technically the Raspberry Pi is an ARM board. You could use and it would probably be a little more relevant.