Geek that I am, I put together a spreadsheet to generate random first-middle combinations from the names we are considering. It's something I've used in the past to make up names for businesses/brands, and I thought it would be useful for this, too.
BE SURE TO MAKE YOUR OWN COPY TO WORK WITH. If you add names to the shared copy, other people will change it around too and nobody will have the names they want to see. Which could be interesting, but probably not very productive. (I left a few names in there just to demonstrate how it works.)
To make a copy:
Click 'File' at the top left of the toolbar
Click 'Make a Copy' in the drop down menu
Give it a name you'll be able to find in your docs
Close out of the shared doc and open your new copy to add your names
To generate your combinations, add names to the first and second columns on the Source List tab. Do NOT leave any empty cells in the middle of your lists or the Combinations tab won't pick up all the names.
Flip to the Combinations tab to see random combinations from your potential names and see what jumps out at you.