Yes, there is such a non-computable set $M$.

Let $M=(M(0),M(1),\ldots)$ be a bi-immune set (i.e., having no infinite computable subset, and whose complement has no infinite computable subset) of minimal Turing degree. (Any nonhyperimmunefree degree contains a biimmune set, so we can use Sacks' minimal degree below $0'$.)
Consider any computably selected sequence
$$
N = (M(f(0)),M(f(1)),\ldots)
$$
Suppose $N$ is computable. Since the range of $f$ is infinite, consider a computable increasing subsequence $f(i_1)<f(i_2)<\ldots$, with $i_1<i_2<\ldots$. Then $\{f(i_j): M(f(i_j))=1\}$ would be an infinite computable subset of $M$ (it's infinite by co-immunity of $M$).

Thus $N$ is noncomputable. Moreover $N\le_T M$. Since $M$ is of minimal degree, it follows that $M\equiv_T N$, i.e., $M$ can be recovered from $N$.