I was trying to explain to a non-pilot why increased power is required
with
altitude. She said "isn't the air thinner up there so there isn't as much
resistance?" I said "yes, but the plane needs to fly fast enough for the
air
over the wings to feel like it does down low. So the speed required goes
up
you get higher. More speed need more power."

This didn't really do the trick.

Can someone think of a better way of putting it without resorting to
mathematics and an explanation of IAS and TAS?

In a word, NO.

It is an issue of physics, and physics uses a lot of math.

To maintain the same TAS, she is right--untill IAS drops to the back side of
the power curve for the altitude at which she is then flying.

To maintain the same IAS, the power requirement will only increase linearly
in proportion to TAS with increasing altitude--until mach number becomes a
consideration (at some significant fraction of unity)

Therefore, within very finite limits, increasing altitude simply allows an
airplane to be flown at a higher TAS while holding the IAS within an
efficient range. That has the effect of only requiring the power to
increase linearly with speed--rather than as the square of the speed
increase.

I hope this helps.

Peter