To clarify, the Wyze Cam Outdoor base station does not have a customer usable factory reset. (I’m thinking the pin hole may be for something they do at the actual factory, otherwise why does it exist?) This article references the camera but it actually applies to both:
Based on my own experience and the postings by some others, this problem appears to specifically related to WCO base stations that were set to wi-fi “passthrough”, meaning they were setup to use wi-fi for the base station’s own network connection and not an ethernet cable.
In this scenario if the base station will not auto-switch back to ethernet when plugged in, even if you try a reboot, your looking at the “blinking blue LED of death”. 
Another sign is that you may be able to switch the unit to travel mode using the sync button on the side of the base station but cannot get it to connect to wi-fi or ethernet. Flashing the firmware also doesn’t help.