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...)
The physicists (see e.g. this paper of Aganagic and Vafa) will write the mirror as a threefold $X$ which is an affine conic bundle over the holomorphic symplectic surface $\mathbb{C}^{\times}\times \mathbb{C}^{\times}$ with discriminant a Seiberg-Witten curve $\Sigma \subset \mathbb{C}^{\times}\times \mathbb{C}^{\times}$. In terms of the affine coordinates $(u,v)$ on $\mathbb{C}^{\times}\times \mathbb{C}^{\times}$, the curve $\Sigma$ is given by the equation
$$
\Sigma : \ u + v + a uv^{-1} + 1 = 0,
$$
and so $X$ is the hypersurface in $\mathbb{C}^{\times}\times \mathbb{C}^{\times} \times \mathbb{C}^2$ given by the equation
$$
X : \ xy = u + v + a uv^{-1} + 1.
$$
From geometric point of view it may be more natural to think of the mirror not as an affine conic fibration over a surface but as an affine fibration by two dimensional quadrics over a curve. The idea will be to start with the Landau-Ginzburg mirror of $\mathbb{P}^{1}$, which is $\mathbb{C}^{\times}$ equipped with the superpotential $w = s + as^{-1}$ and to consider a bundle of affine two dimensional quadrics on $\mathbb{C}^{\times}$ which degenerates along a smooth fiber of the superpotential, e.g. the fiber $w^{-1}(0)$. In this setting the mirror will be a hypersurface in $\mathbb{C}^{\times}\times \mathbb{C}^{3}$ given by the equation
$$
xy - z^2 = s + as^{-1}.
$$
Up to change of variables this is equivalent to the previous picture but it also makes sense in non-toric situations. Presumably one can obtain this way the mirror of a Calabi-Yau which is the total space of a rank two (semistable) vector bundle of canonical determinant on a curve of higher genus.
Best Answer
There is a version of mirror symmetry, called "local mirror symmetry", for certain non-compact Calabi-Yaus, for example the total space of the canonical bundle of P^2 (exercise: show this is CY). The mirror (or rather one possible mirror) of this non-compact Calabi-Yau is an affine elliptic curve in (C^*)^2. I don't think that there is as yet a version of mirror symmetry for more general non-compact CYs, though I don't know too much about this story. In all of the papers that I've looked at on this stuff at least, the only non-compact CYs that have been considered in mirror symmetry so far are total spaces of vector bundles over compact (probably Fano) things.
I guess we should probably get some sort of "Hodge diamond symmetry" in local mirror symmetry, but the story becomes more complicated. One immediate thing to notice is that, at least in the example I've given, the dimensions of the manifolds aren't the same! So things will have to be modified.
The Hodge diamond symmetry in mirror symmetry for compact Calabi-Yaus should really be thought of as coming from a correspondence between certain deformations of a Calabi-Yau and certain deformations of its mirror. In the non-compact case, the deformations that we should consider will be somewhat different from the deformations that we should consider in the compact case.
A somewhat recent point of view, due to Kontsevich, is that this correspondence between deformations can be gotten from homological mirror symmetry. In homological mirror symmetry for compact Calabi-Yau manifolds, we consider a derived category of coherent sheaves on one side and a Fukaya category on the other side. Then we should have an equivalence of categories, plus an equivalence of a certain structure on their Hochschild cohomologies -- in particular their Hochschild cohomologies should be equivalent at least as vector spaces. These Hochschild cohomologies should be thought as the appropriate deformation spaces (or maybe rather the tangent spaces to the appropriate deformation spaces?), and an appropriate identification of the Hochschild cohomologies (plus the extra structure that I mentioned) should give in particular the Hodge diamond symmetry. There should also be homological mirror symmetry for non-compact Calabi-Yau manifolds, but we must define the analogues of derived category and Fukaya category in this situation appropriately. Then there should be an analogous story on the Hochschild cohomologies of the categories, and an appropriate analogue of the Hodge diamond symmetry. See the paper "Hodge theoretic aspects of mirror symmetry" by Katzarkov-Kontsevich-Pantev for more details.