Lattice gas based models are usually discussed in terms of spin averages instead of distribution functions. As they are very useful in the study of adsorption phenomena, a density functional (DF) formalism, which would unify the discussion of both the liquid and the adsorbed phases, seems a most useful alternative. Here we present a first step in that direction by deriving the two essential components needed for any DF theory. The first one is a fully developed Ornstein-Zernike (OZ) formalism which we arrive at in two steps. The first one is the definition (through functional differentiation of the grand canonical partition function) of the distribution and correlation functions hierarchies. In the second step we find that the rigid neighbourhood of a lattice gas forces us, if an authentic DF theory is our goal and even in the grand canonical ensemble, to define N-modified distribution and correlation functions much in the same way as we have recently done when discussing DF theory in the canonical ensemble. These N-modified hierarchies of correlation functions are, indeed, linked by a full set of n-body OZ equations. The second ingredient for any DF theory is an expression for the entropy (in terms of the already discussed correlation functions) which we obtain by following previous work by us in fluids. We also generalize the compressibility contribution to the entropy by using the already derived lattice gas formalism in a way immediately translatable to liquids. In summary, we show how a deep and intimate relationship between lattice gases and fluids can be obtained if both are discussed in a DF framework with functional differentiation techniques and, therefore, we think that the beginnings of a DF theory of lattice gases are established.