Smallest convex set containing given point
Here is an idea to find a "smallish" simplex containing your given point $x$. It does not find a smallest simplex (except by luck).

Construct the convex hull $H$ of your set of $M$ points.
For each $p$ of your set $M$, shoot a ray from $p$
through $x$ until it hits a facet $F$ of $H$. Then $F \cup p$ is a simplex containing $x$. Repeat for all $p \in M$ and retain the smallest volume simplex.

It is easy to create examples where this is terrible, as it is biased toward a simplex
containing a facet of the hull. One can imagine various heuristics to improve, for
example, discarding from $M$ the vertices of $H$ and repeating.