What does the word "symplectic" mean?  I know the definition of symplectic structure, symplectic group, and so on. But what does the word "symplectic" itself mean? 
Meta question: I have many other mathematical words whose etymologies are obscure to me. Is it OK for me to ask one question per such word?
 A: There is a brief explanation here. It looks like the term was coined by Weyl, and was a result of modifying the Latin prefix “com-” from “complex” to the equivalent Greek prefix “sym-”. This is a pretty obscure way to coin a word if you ask me!
A: The term "symplectic group" was suggested  in The Classical Groups: their invariants and representations (1939, p. 165) by Herman Weyl:

The name "complex group" formerly advocated by me in allusion to line complexes, as these are defined by the vanishing of antisymmetric bilinear forms, has become more and more embarrassing through collision with the word "complex" in the connotation of complex number. I therefore propose to replace it by the corresponding Greek adjective "symplectic." Dickson calls the group the "Abelian linear group" in homage to Abel who first studied it.

Take a look at the Earliest Known Uses of Some of the Words of Mathematics web page.
A: The word "sum-plectic" as a greek translation of "com-plexus" was needed also to differentiate the study of "complex geometry" (complex numbers etc) from the study of "complexes de droites" (the geometry of 'line complexes') where the Plücker coordinates are associated to a natural symplectic structure on the space of affines lines in any euclidean space.

The concept of "differential symplectic geometry" has been introduced I believe by J.-M. Souriau in is 1953 paper 
@inproceedings{Sou53,
    Author = {Jean-Marie Souriau},
    Booktitle = {Coll. Int. CNRS},
    Pages = {53},
    Publisher = {CNRS, Strasbourg},
    Title = {G{\'e}om{\'e}trie symplectique diff{\'e}rentielle. Applications.},
    Year = {1953}}
He introduces there the concept of "Variétés isotropes saturées", called today "lagrangian manifolds", name given later by V.-I. Arnold.
A: From Pietro Majer's comments I learn that "symplectic" is the past participle of a classic Greek verb which means "to embrace".
Consequently I would just remark then how surprising is the effectiveness of this adjective to reflect the pervasiveness of the ideas from symplectic geometry in modern mathematics, which interconnect many different subjects.
Infact, from the introduction to the paper "Symplectic Geometry" by A.Weinstein, I quote:

I think it is not unfair to say that symplectic geometry is of interest today, not so much as a theory in itself, but rather because of a series of remarkable "transforms" which connect it with various areas of analysis.

A: The following is from page 1 of *Lectures on symplectic geometry* by Ana Cannas da Silva:

As a curiousity, note that two centuries ago the name symplectic
geometry did not exist. If you consult a major English dictionary,
you are likely to find that symplectic is the name of a bone in a
fish's head. However ... the word symplectic in mathematics was
coined by Weyl who substituted the Latin root in complex by the
corresponding Greek root in order to label the symplectic group. Weyl
thus avoided that this group connote the complex numbers, and also
spared us from much confusion that would have arisen, had the name
remained the former one in honor of Abel: abelian linear group.

