As far as I know, there is no convincing answer for why does the Higgs potential has a 'Mexican hat' shape. Apart from Lorentz invariance and gauge invariance, the Higgs potential of the Standard model has been chosen somewhat arbitrarily.
Do more advanced theories that go beyond the Standard model such as Grand Unified theories, supersymmetry or the theory of inflation have any deeper answer?
Working in natural units ($c=\hbar=1$), each physical quantity has a mass dimension. For the action $S=\int d^4 x \mathcal{L}$ in 4-dimensional spacetime, it's $0$; for $x^\mu$, it's $-1$; for $d^4 x$, it's $-4$; for $\mathcal{L}$, it's 4. A kinetic term $\partial_\mu\phi^\ast\partial^\mu\phi$ in this Lagrangian density implies $\partial_\mu\phi$ is of dimension $2$ while $\partial_\mu$ is of dimension $1$, so $\phi$ is of dimension $1$. Thus if $\lambda_n\phi^n$ is in the potential $\lambda_n$ has mass dimension $4-n$. Renormalisation typically requires this to be $\ge 0$, so we stop at $\phi^4$; this is called scalar $\phi^4$ theory.
Well, technically if we want a $U(1)$ theory for the Higgs, we're limited to $(\phi^\ast\phi)^k$ terms with $k\in\{0,\,1,\,2\}$. As long as neither of the $k>0$ terms has coefficient $\neq 0$, we're done. We can shift the $k=0$ term by an arbitrary constant, so feel free to set it to $0$ or make the potential a perfect-square function of $\phi^\ast\phi$.
You may wish to work out what happens in $d$-dimensional spacetime.
