= Solution
The <Weinstein neighborhood theorem> identifies a neighborhood of $X$ in $M$ with a neighborhood of the zero section in $T^*X$. If $j$ is sufficiently $C^1$-close to the inclusion, projection of its image to $X$ is a diffeomorphism. After reparametrization, $j(X)$ is therefore the graph of a small one-form $\beta$.
The <graph of a closed one-form is Lagrangian> criterion says that $j(X)$ is Lagrangian exactly when $d\beta=0$. Since $H^1(X;\mathbb R)=0$, write $\beta=df$. Intersections of $j(X)$ with $X$ are the critical points of $f$. A smooth function on a compact manifold has a maximum and a minimum; if they coincide as points because $f$ is constant, every point is an intersection. Thus there are at least two intersection points, proving the <nearby exact Lagrangian intersection lemma>.
The cohomology hypothesis is necessary. Take the zero section in $T^*S^1$ and the graph of the arbitrarily small nowhere-zero closed one-form $\varepsilon,d\theta$. Both are Lagrangian and disjoint.
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