Gian-Carlo Rota was perhaps thinking about Matroid theory. His work is cited for example in the preprint "Matroid for algebraic geometers" by Eric Katz.

If Grothendieck has known about lattices, he may have defined a cohomology theory for lattices. Gian-Carlo Rota actually defined homology groups for certain subsets, called cross-cuts, of a lattice in 1964. Whether Grothendieck could have anticipated the numerous connections between matroid theory, tropical geometry, enumerative geometry and algebraic geometry, I don't know.

When it comes to the quote about Grothendieck school clumsily reinventing the rudiments of lattice theory, Garrett Birkhoff's book on lattices (1948 edition) has a section about applications to algebra and algebraic geometry. The well-known fact that "every algebraic variety has a unique expression as an irredundant sum of a finite number of irreducible components" is derived from a general result about distributive lattices satisfying the descending chain condition (ch IX 8.). An algebraic geometer may be able to check if this specific result about lattices applies to some category of schemes and gives a theorem of Grothendieck.