Categorification of Floer homology Floer homology associates a vector space $HF^\ast(L_1,L_2)$ to any pair of Lagrangian submanifolds $L_1,L_2$ inside a symplectic manifold $X$.  By a categorification of Floer homology, I mean a category $\mathcal H\mathcal F(L_1,L_2)$ whose Hochschild homology is isomorphic to $HF^\ast(L_1,L_2)$.
Example: Let $M$ be a closed symplectic manifold, and consider the diagonal $\Delta\subseteq M\times M^-$, which is Lagrangian.  It self-Floer homology $HF^\bullet(\Delta,\Delta)$ is (conjectured to be) isomorphic to the Hochschild homology of the Fukaya category of $M$.  Thus $\mathcal F(M)$ is a categorification $\mathcal H\mathcal F(\Delta,\Delta)$ of $HF^\ast(\Delta,\Delta)$.  Note $HF^\ast(\Delta,\Delta)$ is a unital algebra, and expressing this unit as a Hochschild cycle of $\mathcal F(M)$ is of fundamental importance in the study of the Fukaya category.

Are there any other known circumstances under which a categorification of Floer homology exists?

Floer homology also has a product $HF^\ast(L_1,L_2)\otimes HF^\ast(L_2,L_3)\to HF^\ast(L_1,L_3)$, so it is natural to further ask that a categorification $\mathcal H\mathcal F$ have natural functor $\mathcal H\mathcal F(L_1,L_2)\times\mathcal H\mathcal F(L_2,L_3)\to\mathcal H\mathcal F(L_1,L_3)$ which upon applying Hochschild homology recovers the product on Floer homology.

What is a natural geometric description of a functor $\mathcal F(M)\times\mathcal F(M)\to\mathcal F(M)$ corresponding to the product on $HF^\ast(\Delta,\Delta)$?

 A: I am sort of an armchair mathematician these days, but I suspect the answer to your first question is "no". At least in the hep-th literature categorification of Khovanov homology I have seen many instances of, Floer homology in general I know much less about.  There are just so many:


*

*Floer homology

*Knot Floer homology

*Bordered Floer homology

*Instanton Floer homology

*Monopole Floer homology 

*Heegard-Floer homolo

*etc.


So the theory specializes in many different ways, and they all have really great properties.  However, there are simply too many and I have lost track. 
On the physics side, I know the way to understand all of these is Chern-Simons Theory and a small amount of google-fu returns 


*

*Fivebranes and 3-manifold homology

*Resurgence in complex Chern-Simons theory
I am not an expert in 3-manifolds, so I can't say whether they fit into your question or not.  I know a lot of the mathematical work is all about showing these functors map nicely onto the properties of 3-manifolds so that these define invariants.  
Here I'm just being a librarian and not really opening up the papers in any way.  I can read through Audin's Morse Theory and Floer Homology and notice complexes jump around as you change the choice of height function.  There's a nice course on YouTube by Denis Auroux.
