Why are free objects "free"?
Jakob Nielsen's 1921 paper in Danish, with the title "Om Regning med ikke-kommutative Factorer og den Anvendelse i Gruppetoerien", is where he proved that subgroups of free groups are free. I don't know about the original Danish, but the English translation by Anne W. Neumann is available in Nielsen's collected works, with translated title "On calculation with Non-commutative Factors and its Application to Group Theory", and it contains the following sentences
In our case we will call it the 'free group $G_n$' generated by the generators $a_1,...,a_n$ and will denote it $[a_1,...,a_n]$.
...
In $G_n$, two distinct products of the $a_i$, irreducible in the $a_i$, always describe two different elements. It is for this reason that $G_n$ is called a free group.
One finds free groups described but not named in earlier works, such as Dehn's 1912 paper "Über unendliche discontinuierliche Gruppen", with English translation by John Stilwell titled "On infinite discontinuous groups" in Dehn's collected works:
Consider, say, the group which is given by two generators $S_1$ and $S_2$ without defining relations...
There is a telling passage later in this same paper. Dehn's focus in this paper is to understand groups in which there is a presentation with a single relator, and every generator appears exactly twice in that relator (he proves that this characterizes surface groups). Along the way he considers variations on this property, such as groups in which "each generator appears in the relation either twice or not at all", and then he introduces some terminology:
... we arrive at the following normal form of the group: among the generators are $w$ generators which appear in no relation ("free generators")...
Free objects were first defined* by MacLane in Duality for Groups. That paper gives "free" a curious political context, I quote from page 486:
Call the dual (in this sense) of a free (nonabelian) group a fascist group. R . Baer has shown me a proof of the elegant theorem: every fascist group consists only of the identity element.
You can find a critical discussion of this joke on "freedom versus fascism" at nForum and at MSE.
*As pointed out by Lee Mosher and Peter LeFanu Lumsdaine, free groups were introduced much earlier. MacLane's 1950 paper gave the first definition of a free object in the context of an arrow composition from category theory.