MY ROUTER WILL NOT ALLOW ME TO DELETE AN AMAZON FIRE STICK. I HAVE NEVER OWNED ONE!

> [...] I've tried everything I can think of, [...]

   Think, for a moment, about how little useful information that
non-description conveys to the non-psychics in your audience.

> [...] I am not able to remove this phantom "device" from the list of
> Attached Devices on the router settings.

   "not able" is not a useful problem description.  It does not say what
you did.  It does not say what happened when you did it.  As usual,
showing actual actions (commands) with their actual results (error
messages, LED indicators, ...) can be more helpful than vague
descriptions or interpretations.

> [...] The router I mention was just purchased (refurbished), [...]

   A setings reset is usually a good idea when you get a used device.

   Visit http://netgear.com/support , put in your model number, and look
for Documentation.  Get the User Manual (at least).  Read.  Look for
"Factory Settings" and/or "Reset button".

   Did you configure the wireless-network security stuff (SSID(s),
passphrase(s)) on the new router?  Look for "Specify basic WiFi
settings" in the User Manual.

> [...] the phantom "Amazon Fire Stick" started appearing on the router
> immediately after I switched routers out.

   Some of us can't see that.  What does it say?