Rather than using a mini USB cable to charge the NANO V3.0 Development Board - Clone Compatible Model, I want to use a lithium ion battery pack. Like this one here. How would I connect the two? Would I use a cable adapter between JST and mini USB?
To do what you want, you could indeed use a battery pack with a JST to mini USB cable similar to this one. The tricky part is finding the right cable with a 2 pins JST connector only (since you just want to power up your device).
Moreover, be careful with your battery pack, it's only a 3.7V and you will need at least a 7V battery pack to supply your development board.