Portable 5v power supply to duplicate wall wart
Ok, that 120hz voltage wave means a 'full-wave' rectifier (with no effective capacitor) is being used.

Most likely there is a capacitor inside the device that's actually storing power at the max peak voltage of that waveform.


The device is then (likely) feeding a resistive/linear Vreg from that elevated voltage.
That Vreg is likely dropping your 'real' 5VDC power supply too low to run the circuit's electronics.


DANGEROUS ADVICE: You may have to carefully raise the output voltage of your power supply in order to compensate for the drop.  

Unfortunately, without opening the wifi device &amp; doing at least enough reverse-engineering to figure out what Vmax your device is storing pre-regulator (and preferrably finding the output voltage after it, too) I can't really tell you what voltage to try, or exactly how unsafe any specific voltage would be for your device.