How should I deal with a passenger experiencing hypoxia over the Grand Canyon?
First, let's be crystal clear on something: A person experiencing symptoms of acute hypoxia is an emergency which requires immediate action to ensure the safety of that person. In an in-flight emergency requiring immediate action, the pilot in command may deviate from any rule …  to the extent required to meet that emergency. (FAR 91.3 (b)).

So if you're over the Grand Canyon at 11,500 feet and your mother starts showing signs of hypoxia you key the mic and let ATC know you have a passenger showing signs of hypoxia, tell them you are descending to a lower altitude, and ask for vectors to an airport where you can land. If they give you any trouble about it you respond with "N12345 is declaring a medical emergency".



All that said, there are sensible things you can do to mitigate the risk of hypoxia and avoid the need to declare an emergency.
If you intend to spend an extended period of time at altitude (which I personally define as "Over 10,000 feet" for an average healthy non-smoker):


Bring a portable oxygen system
Ideally with enough cannulas/outlets for yourself and your mother/other passengers).
If a full oxygen system isn't an option bring one or more "boost cans"
These can provide temporary/limited relief for symptoms of hypoxia – probably enough to get yourself back on the ground, or at least to a lower altitude.
Carry a relatively inexpensive pulse oximeter
Check your SpO2 periodically to catch hypoxia early - before clinical symptoms develop.