= Solution
For the graph embedding $\iota(q)=(q,p(q))$, pull back the <symplectic form> in <Darboux coordinates>:
$$
\iota^*\omega=\sum_{i,j}\frac{\partial p_i}{\partial q^j}\,dq^i\wedge dq^j
=\sum_{i<j}\left(\frac{\partial p_i}{\partial q^j}-\frac{\partial p_j}{\partial q^i}\right)dq^i\wedge dq^j.
$$
The coordinate two-forms are linearly independent. The graph has dimension $n$, so it is a <Lagrangian submanifold> exactly when this <pullback> vanishes, equivalently when its displayed coefficient matrix is symmetric.
Let $\theta_L=\sum_i p_i(q)dq^i$. Then $d\theta_L=-\iota^*\omega$, so the same condition says that $\theta_L$ is a <closed differential form>. On a sufficiently small contractible coordinate neighborhood the <Poincaré lemma> supplies $S$ with $\theta_L=dS$. Consequently
$$
\boxed{p_i=\partial_iS.}
$$
One explicit primitive on a star-shaped neighborhood of $q_0$ is $S(q)=S(q_0)+\int_0^1p_i(q_0+t(q-q_0))(q^i-q_0^i)\,dt$. Differentiate it and use $\partial_jp_i=\partial_ip_j$ to recognize the derivative of $t p_j(q_0+t(q-q_0))$; the result is $\partial_jS=p_j(q)$. Conversely $p=dS$ has symmetric mixed derivatives and gives a <Lagrangian graph>. This construction requires the local projection to the $q$ coordinates to be nonsingular. A general <Lagrangian submanifold> need not be such a graph in a preassigned chart, and a closed one-form on a larger domain need not be globally exact.
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