I would not put lattices (in the sense of universal algebra) in a role that was "central" to all of mathematics.  Nor would I do that with a Denjoy Integral, or distributions, generating functions, nor even sets nor categories.  All of those are tools or ways of thinking that are frequently being adapted or related to various other parts of mathematics.  The interplay of the various notions is a reason (whether conscious or unconscious varies among individuals) to study and develop mathematics, in my not so humble opinion.

I would say that lattice theory is essential to the fields of logic and algebra that I have studied. Galois connections between model classes and certain deductively closed sets of theories resulting in lattices of varieties and other classes, algebraic properties resulting from the characteristics of congruence lattices of a class of algebras, the classification using tame congruence theory of finite algebras into five types, forms of algebraic logic inspired by Heyting and other lattice-like algebras, these alone are reason enough for many to study the subject.

When we go to Dedekind-MacNeille completions, realizing that many inequalities involving max and min are lattice operations in another guise, and applying lattice theory in various ways to number theory, partial differential equations, computational complexity, and seeing how such a concept is useful to one's mathematical sphere, then one can say (or not) "Lattices are central to MY mathematics."

Gerhard "Mathematics For One, For All" Paseman, 2013.12.18