The existence certainly remains true if the ambient metric is real-analytic, and this follows from the Cartan-Kähler Theorem since all you are asking for is local minimal surfaces.
In the case of a smooth metric, because the minimal surface equation is determined elliptic, the existence of a local solution attaining a specified $k$-jet satisfying the equation follows from standard elliptic theory (and you are only asking to specify the $2$-jet of the minimal surface at a point). (For example, see the discussion on elliptic equations in the appendix to Besse's Einstein Manifolds.
While Igor Rivin's answer does suggest that there ought to be many solutions attaining a given $k$-jet, this does not follow immediately from the existence of solutions to the boundary value problem. At least, I do not see how to prove it without invoking elliptic local solvability, in which case you might as well start with that.
The higher dimensional case is not significantly different: In the smooth case, one can specify the $k$-jet of a minimal surface at a point for any $k\ge 1$ as long as the specified $k$-jet satisfies the constraints of the minimal surface equation. (Again, this follows from the Cartan-Kähler Theorem for real-analytic metrics and standard elliptic theory in the smooth case, see Besse, as above.)