This is a great question I wish I understood the answer to better.
I know two vague answers, one based on derived algebraic geometry and one based on string theory.
The first answer, that Costello explained to me and I most likely misrepeat,
is the following. The B-model on a CY X as an extended TFT can be defined in terms of
DAG: we consider the worldsheet $\Sigma$ as merely a topological space or simplicial set (this is a reflection of the lack of instanton corrections in the B-model), and consider the mapping space $X^\Sigma$ in the DAG sense. For example for $\Sigma=S^1$ this is the derived loop space (odd tangent bundle) of $X$.. In this language it's very easy to say what the theory assigns to 0- and 1-manifolds: to a point we assign coherent sheaves on $X$, to a 1-manifold cobordism we assign the functor given by push-pull of sheaves between obvious maps of mapping spaces (see e.g. the last section here). For example for $S^1$ we recover Hochschild homology of $X$. Now for 2-manifold bordisms we want to define natural operations by push-pull of functions, but for that we need a measure -- and the claim is the Calabi-Yau structure (together with the appropriate DAG version of Grothendieck-Serre duality, which Kevin said Lurie provides) gives exactly this integration...
Anyway that gives a tentative answer to your question: the B-model assigns to a surface $\Sigma$ the "volume" of the mapping space $X^\Sigma$, defined in terms of the CY form.
More concretely, you chop up $\Sigma$ into pieces, and use the natural operations on Hochschild homology, such as trace pairing and identification with Hochschild cohomology (and hence pair-of-pants multiplication).. of course this last sentence is just saying "use the Frobenius algebra structure on what you assigned to the circle" so doesn't really address your question - the key is to interpret the volume of $X^\Sigma$ correctly.
The second answer from string theory says that while genus 0 defines a Frobenius manifold you shouldn't consider other genera individually, but as a generating series -- i.e. the genus is paired with the (topological) string coupling constant, and together defines a single object, the topological string partition function, which you should try to interpret rather than term by term. (This is also the topic of Costello's paper on the partition function). BTW for genus one there is a concrete answer in terms of Ray-Singer torsion, but I don't think that extends obviously to higher genus.
As to how to interpret it, that's the topic of the famous BCOV paper - i.e. the Kodaira-Spencer theory of gravity. For one thing, the partition function is determined recursively by the holomorphic anomaly equation, though I don't understand that as "explaining" the higher genus contributions. But in any case there's a Chern-Simons type theory quantizing the deformation theory of the Calabi-Yau, built out of the Kodaira-Spencer dgla in a simple looking way, and that's what the B-model is calculating.
A very inspiring POV on this is due to Witten, who interprets the entire partition function as the wave function in a standard geometric quantization picture for the middle cohomology of the CY (or more suggestively, of the moduli of CYs). This is also behind the Givental quantization formalism for the higher genus A-model, where the issue is not defining the invariants
but finding a way to calculate them.
Anyway I don't know a totally satisfactory mathematical formalism for the meaning of this partition function (and have tried to get it from many people), so would love to hear any thoughts. But the strong message from physics is that we should try to interpret this entire partition function - in particular it is this function which appears in a million different guises under various dualities (eg in gauge theory, as solution to quantum integrable systems, etc etc...)
Since no one else has tried to answer, I'll take a shot. It seems to me that there are threads of ideas in this story that in the very distant future might be woven together to give a possible answer.
To begin, we should note that there seems to be a general idea, discussed in this mathoverflow question, Deformation quantization and quantum cohomology (or Fukaya category) -- are they related?. That one could define the Fukaya category as modules over a deformation quantization of $C^{\infty}(X)$ corresponding to the symplectic form $\omega$.
The basic idea is that in two naive respects this category of modules behaves a lot like the Fukaya category. Firstly, the Hochschild cohomology of the deformation quantization is almost by definition the Poisson cohomology of the symplectic form $\omega$, which in turn is known to be isomorphic to $H^*(X)((t))$. As an equation:
$$HH^*(A_\omega,A_\omega) \cong H^*(X)((t)) $$
Second, one can define a reasonable notion of modules with support on a Lagrangian submanifold and for any Lagrangian L, produce canonical holonomic modules supported there. One can compute that $$Ext(M_L,M_L) \cong H^*(L)((t))$$ There is some hope that one can put in the instanton corrections in a formal algebraic way and a fair amount of work has been done in this direction.
This story works best so far for the Fukaya category of $T^*X$ where the deformation quantization is roughly the algebra of differential operators. This is related to more work than I could competently summarize. I'll just mention, work of Nadler and Zaslow, Tsygan and Tamarkin. This approach is used by Kapustin and Witten to incorporate co-isotropic branes into the Fukaya category in their famous study of the Geometric Langlands. There, they are after some enlargement of Nadler's infinitesimal Fukaya category of $T^*(X)$. Note however that this not the same Fukaya category(the wrapped Fukaya category) that one studies in the context of mirror symmetry, but perhaps things will work better in the compact case if that is ever put on firm ground.
This was all a prelude to say that deformation quantization places you firmly in the land of non-commutative geometry anyways. Things like differential operators for non-commutative rings can make sense http://www.springerlink.com/content/r0rqguawu1960qxy/. I've never really looked at Van Den Bergh's work, but perhaps the passage from the sheaf of algebraic functions to the sheaf $C^\infty(X)$ is another stumbling point. One of Maxim's Kontsevich's ideas (see his Lefschetz lecture notes http://www.ihes.fr/~maxim/TEXTS/Kontsevich-Lefschetz-Notes.pdf) is that for any saturated dg-algebra there should maybe exist some nuclear algebra which bears the same formal relationship as the algebra of algebraic functions and smooth functions.
Best Answer
Let me summarize and supplement my remarks above. A quaternion-Kahler manifold $X$ is a Riemannian manifold with holonomy $Sp(n)Sp(1)$, its definition of course includes hyperkahler manifolds as a special case. For a hyperkahler manifold, mirror symmetry can sometimes be realized as a hyperkahler rotation, e.g. elliptic $K3$ surfaces. However, such an understanding fails in general, as was pointed out by Huybrechts in his lecture notes: http://arxiv.org/pdf/math/0210219.pdf. For the method to handle mirror symmetry in the general case, see the work of Gross-Siebert: http://arxiv.org/abs/math/0703822.
Now let's exclude hyperkahler manifolds by assuming the scalar curvature is nonzero. For simplicity, let's further assume that $X$ is symmetric, then it can be written as $X=G/H$ with $G$ a simple Lie group and $H=K\cdot SU(2)$ , $K$ is the centralizer of $SU(2)$ in $G$. In this case, as far as I konw, only the mirror symmetry for the case when $G=SU(n+2)$ and $H=S\big(U(n)\times U(2)\big)$ is systematically studied.
In general, for the homogeneous space $X=G/P$, where $G$ is a semisimple Lie group and $P$ a parabolic subgroup, its mirror is given by the Landau-Ginzburg model $(R,W)$, where $R\subset G^L/P^L$ is the Richardson variety, and $W:R\rightarrow\mathbb{C}$ is a holomorphic function called superpotential. Here $G^L$ and $P^L$ denote their Langlands dual. The main reason for using $X^\vee=R$ to partially compactify $(\mathbb{C}^\ast)^N$ is to get the following isomorphism between the quantum cohomology ring of $X$ and the Jacobi ring of $W$:
$QH^\ast(X)\cong Jac(W)$.
This isomorphism is generally expected to hold for mirror symmetry for Fano manifolds. For the case when $X$ is toric Fano, such an isomorphism is proved by Fukaya-Oh-Ohta-Ono: http://projecteuclid.org/euclid.dmj/1262271306. The general mirror construction for $X=G/P$ is done by Rietsch: http://arxiv.org/abs/math/0511124.
The isomorphism $QH^\ast(X)\cong Jac(W)$ should be compared with Kontsevich's conjecture, which asserts the following:
$QH^\ast(X)\cong HH^\ast\big(\mathcal{F}(X)\big)$,
where $\mathcal{F}(X)$ is the Fukaya category of $X$. The interesting thing is that when $X$ is Grassmannian (e.g. $X=Gr(2,n+2)$, in which case $X$ is quaternion-Kahler), $W:X^\vee\rightarrow\mathbb{C}$ has only isolated critical points, therefore it should provide a Lefschetz fibration on $X^\vee$, so Seidel's theory (http://www.ems-ph.org/books/book.php?proj_nr=12) can be applied and one may expect that all the geometric information of the Fukaya category $\mathcal{F}(X^\vee,W)$ is contained in the superpotential $W$. This suggests that $\mathcal{F}(X)$ and $\mathcal{F}(X^\vee,W)$ should be related to each other.
From an SYZ point of view, there is no such thing as mirror symmetry for Fano manifolds. Because everytime we talk about mirror symmetry for a Fano manifold $X$ we regard it as a Calabi-Yau manifold $X_0$ together with some boundary divisor $D$, i.e. $X_0=X\setminus D$. The Landau-Ginzburg mirror manifold $X^\vee$ is constructed from $X$, and $D$ determines only the superpotential $W$. For the SYZ mirror construction for some of the Fano examples, see Auroux's fundamental work: http://arxiv.org/abs/0706.3207.
In fact, such a philosophy coming from SYZ mirror symmetry applies also to the case when $X=G/P$. There is a Richardson variety $R^\vee$ which is dual to $R$ inside $G/P$. The open part $R^\vee\subset X$ should be regarded to be SYZ mirror to $R\subset G^L/P^L$. But when we add back the boundary divisor $X\setminus R^\vee$, we need to use the superpotantial $W$ to correct $R$, so we end up with the Landau-Ginzbug model $(R,W)$. From such a point of view, the meaning of the following mirror dualities should be clear:
$R^\vee\leftrightarrow R$
$(R,W)\leftrightarrow G/P$
$G^L/P^L\leftrightarrow (R^\vee,W^\vee)$
Let me also mention that the philosophy that $T$-duality should be realized as the Langlands duality in some special cases dates back to the work of Hausel and Thaddeus on mirror symmetry for Hitchin moduli spaces: http://arxiv.org/pdf/math/0205236v1.pdf. As the quaternion-Kahler case (with non-zero Ricci curvature) should be regarded as parallel to the hyperkahler case, it's not strange that the same philosophy works here.
To get a deep understanding of mirror symmetry in the quaternion-Kahler case, we will need to study the geometric structures of a quaternion-Kahler manifold, and understand how the geometric structures exchanges when passing from $X$ to $X^\vee$. Unfortunately, personally I don't know much development in these directions. However, there is the paper of Leung (http://arxiv.org/pdf/math/0303153v1.pdf) which aims to use normed division algebras to unify the geometric structures coming from different holonomy groups. I personally believe this unified picture should be fundamental in the study of mirror symmetry for quaternion-Kahler case.