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.