Here is a sketch of a proof: Let $\dot f$ be a name for $f$. Wlog we may assume that $\dot f$ is a nice name and hence uses only countably many conditions. It follows that it is an $Fn(J,2)$-name for some countable set $J\subseteq I$. Let $(p_n)_{n\in\omega}$ be an enumeration of $Fn(J,2)$.
We construct a function $g\in M$ as follows. For each $n\in\omega$ choose $g(n)$ such that for all $k\leq n$ the following holds: if $p_k\Vdash\dot f(n)=m$ for some $m\in\omega$, then $g(n)\geq m$.