Abraham Robinson explicitly referred to Leibniz's idea of infinitesimal quantities when developing non-standard analysis in 1960's. Wikipedia article has a quotation from his book Robinson, Abraham (1996). Non-standard analysis (Revised edition ed.). Princeton University Press. ISBN 0-691-04490-2.
Added: the idea of expressing logic in an algebraic way is credited to Leibniz; see e.g. the following article in Stanford Encyclopedia of Philosophy:
Added: Saul Kripke introduced a semantics of possible worlds (really, relational semantics) for modal logic. http://en.wikipedia.org/wiki/Modal_logic#Semantics
The idea of possible worlds precedes Leibniz, but he devoted a lot of consideration to it. Ironically, his claim that our existing world is the best out of possible ones is perhaps most known from the ridicule it received in Voltaire's "Candide". Oh wait, this is Math Overflow...