The three criteria for a USB stick to appear in the dropdown picklist are as follows:
- mountable
- removable
- writable
That third one is important.
Sometimes the OS doesn't see specific USB sticks as removable. There's an easy way to check both of these. Right-click or control-click on the USB stick on your desktop and select "Get info." Is the icon orange or white? White indicates it's removable. You can also check if it's writable at the bottom of the info window.