How to select parameters for elliptic curves not found in standards (Hessian, Jacobi Intersection, Jacobi Quartic, etc)?
Weierstrass models can be converted to Hessian and Jacobi models (where such models exist) and in the days before Edwards curves were considered for ECC-based on implementation and side-channel properties.
The conversion to Jacobi model seems to be covered on mathoverflow.
To convert to Hessian, compute the $j$ invariant in $\mathbb F_p$ and then solve
$$j=\frac{c^3(c^3+216)}{(c^9-81c^6+2187c^3-19683)}$$
for $c\in\mathbb F_p$ (if the corresponding degree 9 polynomial has a root in $\mathbb F_p$). Your curve is then birationally equivalent to
$$X^3+Y^3+Z^3=cXYZ.$$
The Handbook of Elliptic and Hyperelliptic Curve Cryptography section 13.1.5.b shows how to explicitly calculate the binational maps if you want to find the image of specific points on the Hessian.
Thus you could take a standard Weierstrass model (e.g. NIST P256, Curve25519) and likely convert it to an alternative form if you so wished.