Which Arduino nano is best for creating a nano drone? Will it be compatible with 3.7 v coreless brushed motors?
Assuming that your goal here is to make this a hobby project and are not expecting it to be a plug-and-play toy, totally! Like is demonstrated in the tutorial you link to, it is entirely possible to use an Arduino as a basic multirotor flight controller.
However, there will always be external components you'll need to connect to the Arduino/MCU board (excluding exotic boards that come with large/fancy shields) including:
- Some kind of control link receiver (the tutorial uses a Bluetooth breakout)
- Some kind of motor driver (the tutorial appears to use TO-92 BJT transistors, an odd choice...)
- IMU/INS of some kind (the tutorial uses a GPS breakout, another odd choice...)
- Other components I'm probably forgetting
Because of this, most all (small) Arduino boards will work here. The Arduino/MCU only interacts with the motors/battery voltage through the motor drivers, so it shouldn't matter what battery voltage or motor size you use. (at least as it pertains to selecting an MCU)
More powerful/faster MCUs are necessary to implement complicated flight controls and algorithms to improve flight quality, but it's unlikely that you'll be doing that with your first DIY drone. An Arduino Nano should be fine.
NOTE: Generally, I think the aforementioned text tutorial isn't using a very good set of parts. In particular, the motor drivers (should probably be a little more beefy/well-cooled) and INS board (should almost certainly be an Accelerometer/Gyroscope/IMU and not a GPS) don't inspire confidence. Despite not being a nano-scale drone, this tutorial video from Joop Brokking is significantly more grounded in multirotor design/control fundamentals. You could likely adapt his project to a smaller drone frame/motors/etc.
I think you'd want the "Nano 33 BLE" - not that you need Bluetooth Low Energy, but because it has gyro/accelerometers built in. Otherwise you'd need to add them as a separate board, like the GY-521
You could even go for the 'Sense' version to gain a barometer, which you could then use for an approximate altitude-hold function.
You will still need an external board to control the motors, as the Nano can only manage a few milliamps. The motor carrier board you link to looks good, although I suspect the '500amp' output current is a typo for 'milliamps'!
If you get the Arduino with headers, it should plug straight into the motor carrier.
You'll still need some form of radio receiver.
It's worth mentioning that the result will be significantly heavier than a board that is designed for a nano-quad, and could be pushing the limits of brushed motors. You might be better to use a purpose made 4-in-1 ESC board and brushless motors instead.