Its easier to forget about the constructions for questions like this, and look at the formal properties of perverse sheaves. In particular, we know that (in all these circumstances), they form an abelian category $Per(X)$, and that this category has an action of the Verdier duality functor $\mathbb{D}$, a contravariant equivalence. We built this category as the heart of a $t$ structure, but lets forget about the ambient surrounds for a moment.
The simple objects in this abelian category are our IC sheaves, indexed by strata + local system, and since we are in an abelian category, these are characterised internally to $Per(X)$ as the objects with no nontrivial subobjects, or quotients. In particular, since $\mathbb{D}$ is an equivalence, it is exact, so any simple object must map to a simple object (subobjects of $\mathscr{F}$ would give quotients of $\mathbb{D}\mathscr{F}$, etc), giving that it preserves the class of $IC$ sheaves.
So now to work out the $IC$ sheaf corresponding to $\mathbb{D}IC(X_\lambda,\mathscr{L})$, we should note that $\mathbb{D}$ preserves the closure of the support of any (complex of) sheaves, so $\mathbb{D}IC(X_\lambda,\mathscr{L})$ will be supported on $X_\lambda$, and will be simple, so will be of the form $IC(X_\lambda,\mathscr{L}^*)$ for some irreducible local system $\mathscr{L}^*$ on $X_\lambda$. To see which local system $\mathscr{L}^*$ is, since its a pushforward from $\overline{X_\lambda}$, we can check on the smooth locus of $\overline{X_\lambda}$, so we can assume $\overline{X_\lambda}=X$. Now we can just check what the duality functor does on a smooth variety (with trivial stratification), and it follows from the explicit description of the dualising sheaf in this case that its just taking the dual local system, the dual representation of the associated representation of the fundamental group.
So to sum up, $\mathbb{D}IC(X_\lambda,\mathscr{L})=IC(X_\lambda,\mathscr{L}^*)$, where $\mathscr{L}^*$ is the dual local system on $X_\lambda$, and to prove this, one can use functoriality/abelian category nonsense to reduce to the case of the trivial stratification on a smooth variety, for which this is a direct computation.
