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:
http://plato.stanford.edu/entries/leibniz-logic-influence/#DisLeiMatLog
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...
I don't know of a systematic change in notation, but Chern definitely used a different convention than what most other differential geometers did. The decision stems from homogeneous spaces and deciding whether you want to work with left or right cosets. Most people use left cosets, which has the consequence that the action of the isotropy group on the fiber on the principal bundle of orthonormal frames is a right action. Chern chose to do it the other way around (just as Herstein does in his algebra text).
It should be noted that there is no standard notation used by everybody in differential geometry. We all have our individual preferences. My experience is that no matter what you do, you run into situations where your choice of notation doesn't work so well.
This is probably a false memory, but I remember my advisor asking me if I had created my own notation yet.
Someone who has really given this a lot of thought and designed an idiosyncratic but quite versatile notation is Roger Penrose.
My advice is to set up your own preferred conventions and learn to convert whatever notation is being used in a paper or text into yours. It is also good to be comfortable with working with either vector fields directly (which for me is easier to interpret geometrically) or differential forms (where calculations are often a lot easier). And although it is usually better to work with respect to an orthonormal frame of tangent vectors and the corresponding dual frame, it is also useful to be to work with arbitrary frames. And, if you want to do PDE estimates, you usually need to convert everything into local co-ordinates.
ADDED: In terms of explicit formulas for a matrix group, there are two possible choices for the Maurer-Cartan form, either $A^{-1}\,dA$ or $dA\,A^{-1}$, depending on whether you want to work with left or right invariant vector fields. Again, most people use left invariant vector fields and set $\Omega = A^{-1}\,dA$, so the Maurer-Cartan equations are
$$
d\Omega + \Omega\wedge\Omega = 0.
$$
If instead you define $\Omega = dA\,A^{-1}$, you get
$$
d\Omega - \Omega\wedge\Omega = 0.
$$
Best Answer
Let V be the vertex of a parabola, F its focus, X a point on its symmetry axis, and A a point on the parabola such that AX is orthogonal to VX. It was well within the power of the Greeks to prove relations such as $VX:XA = XA:4VF$. If you introduce coordinate axes, set $x = VX$, $y = XA$ and $p = VF$, you get $y^2 = 4px$, the modern form of the equation of a parabola.
Everything now depends on what "invention of coordinate geometry" means to you. I do not think that the Greeks' work on conics should be confused with coordinate geometry since they did not regard the lengths occurring above as coordinates. It's just that parts of their results are very easily translated into modern language.
In a similar vein, Eudoxos and Archimedes already were close to modern ideas behind integration, but they did not invent calculus. Euclid, despite Heath's claim to the contrary, did not state and prove unique factorization. And Euler, although he knew the product formula for sums of four squares, did not invent quaternions (Blaschke once claimed he did). In any case, we are much more careful now with sweeping claims such as "Appolonius knew coordinate geometry" than historians were, say, 100 years ago.