The cable modem obtains a dynamic IP address from the cable company and then passes it to whatever it is connected to, either a PC directly or the WAN side of a router. Some cable modems have router capability built in, so hopefully if it does that's disabled so you aren't doing double NAT translation (which should still work, but it's another potential layer of problems). So, first make sure the router is getting a routable IP (not an internal one like 192.168.1.1) from the modem.

The next time everything stops working, first see that there's no obvious problems on the cable modem (ie its LEDs say it's online and maybe there's some blinking going on to indicate traffic). Then see if you can get to the router's administration page and see what IT says about everything. Does it have an IP? Does it indicate any issues? If everything looks OK, does releasing the IP and renewing it make everything work again?

Of course if you CAN'T get to the administration page, then it's time for a new router. Or if the cable modem indicates a problem and/or things only start working when the modem itself is reset... then it's probably a cable problem (bad line quality or a misconfiguration somewhere).

(Also as an aside, some cable modems have web interfaces you can access, ie. my Motorola SB5100, where you can see online status, signal levels, as well as a log of any errors, although sometimes the cable companies disable this or there may not be one at all.)

I've heard some stories about certain routers handing leases/renewals in a way that the cable company's equipment doesn't like... what's coming to mind was one that was always renewing itself to the point where it was eventually ignored.

If the client just wants something cheap, the Linux-based Linksys one should be good. Myself, I use a Zyxel Zywall 35 that has been utterly reliable (and the *original* Linux-based WRT54GS for wireless).