Technicolor CGM4141 reset button..what should it look like?

Her description of it is that it's not recessed, but protrudes above the surface of the case. It is definitely located where the reset switch is, below the coax connector.

How can we access the modem settings?

I don't think a modem has settings but this particular modem is also a wireless router.  You can go to the following link and navigate to the Personalize Your Wifi Experience section for instructions to manage the router.

If she definitely found the reset button and its sticking out, it's probably broken and may not work anyway.  She could fully depress it until it stops inside the chassis to attempt a reset.

www.cox.com/.../pw6_3.1_pano_wifi.pdf

If it's broken, it's no big deal because she's renting it anyway.  Just go to the Solution Store and freely exchange it.

The option to default the modem to its factory settings is in the menu of the modem?

Are you distinguishing between Cox sending a reset signal to the modem and a factory reset of the modem?

I agree that would work, except that her iPad isn't connected to wifi and the internet, because of this problem. The straightforward way out is to reset the modem so the iPad can connect to it. But there is the problem with the reset button. 

I have the idea that this is a refurb unit that someone else had. The previous owner plugged up the hole, to keep their kid from resetting the modem and taking it out of parental control, and then it got returned and sent to another customer. Without taking the plug out.

Is this the first time trying to connect the iPad to the router?

Is she connecting via 2.4 GHz?

Is the router broadcasting on 2.4 GHz?

Tell her to log into the router...as per instructions above...and verify if 2.4 GHz is enabled.

A factory reset is kinda extreme...don't you think?