Motorola router keeps flashing on and off

Mototola left the router/modem business years ago, and all of the Motorola equipment is EOL at this point. Some ISPs still accept it, but they’re all beginning to drop it now. You’ll need to base if you want to repair it based on if it’s EOL or still supported by your ISP and others in your area, and the odds. I’d give the ISP a call and check first, unless it's super old like DOCSIS 1.0/2.0, which are basically completely unsupported at this point.

If you do fix it, power issues like this are often due to worn out power supplies or bad capacitors in the equipment. Start with the power supply and see if that helps; if not, verify the generation of the equipment and make a determination if it's worth fixing. If it's DOCSIS 1.0/2.0, recycle it. DOCSIS 3.0 may be, in some cases -- as long as the cost is significantly cheaper then a new Arris one.

Depending on your ISP, you may find your money is better spent on a DOCSIS 3.1 modem, which will be supported for a long time and will not be obsoleted in the foreseeable future as some ISPs are in the process. This is generally what ISPs lease now, so it's probably coming any day where you have no choice.