In the Wikipedia article about the tautochrone curve, there is a proof of the fact that the tautochrone curve must be a cycloid. The proof starts with the following statement:
One way the curve can be an isochrone is if the Lagrangian is that of a simple harmonic oscillator: the height of the curve must be proportional to the arclength squared.
How is the statement in bold justified?
The statement in bold is better phrased as:
The motivation for this assumption is the fact that in simple harmonic motion, the period is independent of the amplitude of the motion. So if $s(t)$ could be arranged to follow SHM, the particle's time of descent to its lowest level would be the same regardless of its starting position -- which is the goal of a tautochrone.
To understand what's going on in the rest of that paragraph, recall that the Lagrangian under the SHM assumption would then be $$ L(s) = \text{kinetic energy - potential energy} = \frac12 m\dot s^2 -\frac12 ks^2. $$ This explains why the paragraph is supposing that $y=s^2$ ("the height of the curve must be proportional to the arclength squared"), since the particle's potential energy is proportional to its vertical displacement above the lowest level, which is what $y$ represents. The rest of that paragraph goes on to derive the cycloid solution from the assumption $y=s^2$.