In chapter 1, of Landau-Lifshitz Mechanics' book, Landau through isotropy and homogeneity of space and homogeneity of time proves that the Lagrangian must depend of magnitude of velocity of the particle.
This seemed fine. But he didn't give a reason as to why it must depend of $|v|^2$ and not on some $|v|^n$ where $n\neq2$.
The choice of the dependence is largely arbitrary at that point. In that chapter they just choose lagrangian as $L(v^2)$. If they chose it as $L^\prime(|v|^n)$ in equation $(3.1)$, they'd just have to say $L^\prime(|v|^n)=\frac12 m\sqrt[n]{|v|^n}^2$ in equation $(4.1)$.
Also, this choice is simple enough because it's merely a dot product of velocity with itself, which is the simplest scalar function of a vector.
Firstly, this is the simplest possible dependence. Second and more important this allows to reproduce the second Newton's law from variation of the action $$m \dot{v}=-\frac{\partial V}{\partial x }.$$ Where $V(x)$ is potential in which the particle is moving.
