Hi,
So, powering mobile Arduino projects is an interesting topic on its own. Unfortunately we don't have our own video on it yet, but I can try to explain here.
Most Arduino boards can be powered several different ways. Using the Arduino UNO (the most common/basic board, and what most of our tutorials are for) as an example
https://store.arduino.cc/products/arduino-uno-rev3, there are three ways to power it:
1. Through the USB port from a computer or USB wall charger. USB ports are already at 5 volts, so this provides 5V directly to the Arduino.
2. Through the "barrel jack" connector (the cylindrical plug). This plug will accept an "unregulated" voltage between 7-12V. This lets it work with a range of different wall adapters, or a 9V battery with a barrel jack adapter like this:
https://www.sparkfun.com/products/9518. This voltage then goes through a "voltage regulator" to produce exactly 5V on the Arduino's 5V pin.
3. Through the "Vin" and GND pins directly. Again this will accept a voltage in the range of 7-12V, so for example you could use a 9V battery with this type of adapter:
https://www.amazon.com/Battery-Connecto ... 08SL9X2YC/ to plug it directly into the Arduino pins instead of using a bulkier barrel jack.
Now moving on to the Arduino Nano:
https://store-usa.arduino.cc/products/a ... edStore=us
It does not have a barrel jack connector. So your options to power it are either the mini-USB port (which would already be 5V assuming it's coming from a USB source) or the Vin/GND pins, which would need to be 7-12V. You are correct that a 4xAA battery pack would give you 6V, so that's not enough for the Vin pin. You would need at least 5 or 6 AA batteries which would get pretty bulky. That leaves some other options, two of which you've already mentioned:
1. Coin cell batteries. I did a quick search and have also not had any luck finding a triple coin cell holder for 3x3V = 9V. However, I see a bunch of double holders like this:
https://www.amazon.com/LAMPVPATH-cr2032 ... 07BXDHT4B/. You could wire two of those in series to get 4x3V = 12V total, then power the Nano through the Vin pin. One thing to be careful about here is that coin cell batteries are generally not designed to provide a lot of *current*. When picking a battery, you have to think about both the voltage, and the current required by the electrical load (in this case, the solenoid). In general, things like motors or solenoids use a lot of current. Coin cell batteries are more intended for providing a low amount of current over a longer period of time, for something like a watch or hearing aid. So they might work as a proof of concept here, but might not last as long as other battery options. That leads us to option #2.
2. Lithium batteries. These are generally pretty flat and compact so they are used a lot in wearable electronics projects. They can also provide a lot of power for their size, so they are used in things like drones where weight matters a lot. They are available in many different sizes:
https://www.sparkfun.com/search/results ... um+battery. In general they only produce about 3.7V, so you would need to combine two or three of them in series, then again you could power the Nano through the Vin pin. If you search online you can find packs where the batteries are already combined for you to produce a higher voltage. Note that these batteries are rechargeable, so you need to purchase a compatible charger.
3. 9V battery and the Vin pins. This may be the simplest route to at least get things working initially. Depending on their size, if you have to combine two lithium batteries they could wind up being larger than the 9V battery anyway. But since they're flatter, you could put them side by side instead of on top each other, and then maybe they would fit into the wallet better.
4. Alternatively, you could use a USB power bank:
https://www.amazon.com/s?k=USB+power+bank, also called a portable USB battery etc:
https://www.amazon.com/s?k=portable+USB+battery. These are usually intended for use with phones and vary in size, but you could look for a small one that would fit in a wallet. The only problem here is that most newer phones have USB-C ports, so you'd need to find one that lets you connect a mini-USB cable for the Nano, or get an adapter. This would allow you to power it via the USB port instead of the Vin pins.
Ultimately - you should be able to get the rest of the project working as a proof of concept with any of these options. Evaluating which option is *best* for the final product would be a great addition to the project as part of the engineering design process.
Hope that helps!