What is the reason for the name *left* coset? Let $G$ be a group and let $H \leq G$ be a subgroup. It seems that it is now standard to call the cosets $$gH=\\{gh \ | h \in H \\}$$ the _left cosets_ of $H$ in $G$. I have to admit to being slightly annoyed with this convention: these are the orbits for the _right_ action of $H$ on $G$. Therefore I am perpetually tempted to refer to them as right cosets. Is there a second good reason (I happily admit convention is a very good reason) for calling these left cosets? (The fact that the "g" is written on the left does not count, as far as I am concerned).

Don't think of them as the orbits of the right action of $H$ on $G$. The set of left cosets itself admits a natural left $G$-action, and every transitive left $G$-action arises in this way.