Is this just for fun or are you expecting it to actually accomplish a useful amount of mining?
If it's the former, go for it I guess? If it's the latter, you're many many orders of magnitude off in terms of hardware requirements. Exactly how many orders of magnitude depends on the coin, I guess, but considering that some coins are barely worth mining on a high end GPU capable of tens trillions of operations per second at 32 bits(?), and the MCU in an arduino nano is capable of about 20 million operations per second (depending on what the operation is, some instructions are slower than others) at 8 bits, you're off by something like a factor of 40,000,000.
EDIT: oh, wait "Duino Coin" is a thing? I thought you were looking to mine one of the more high profile coins that's worth real money. I guess if it's meant to work on an arduino then I guess it should work on any arduino compatible hardware, and probably doesn't require any peripherals so just get whatever looks good or whatever kit comes with what you might need for any future projects, it doesn't seem like it really matters