The order is alphabetical by default, and EmulationStation shows them in the order they appear in the configuration file (es_systems.cfg). You can create your own es_systems.cfg and re-arrange them in the file as you want - see https://retropie.org.uk/docs/EmulationStation/#es_systemscfg-edits.

@FlyingTomahawk the script and instructions have been updated on github under my original repo. Here is the link:
github Change_es_systems

I am going to start a new thread with this script and instructions as well.

Edit: link to new thread

@mediamogul I didn't know this. Thank you.

I am using an iPac2 with a real arcade cabinet so I will have to experiment with the button combos to figure it out.

Thanks,
BC