A footnote of Enderton's Elements of Set Theory (1977, page 4) for the definition of power set states that
the reasons for using the word "power" in this context are not very convincing, but the usage is now well established.
I wonder what the not-so-convincing reasons were.
EDIT 1: A question regarding the etymology of power is here.
EDIT 2: On page 141, Enderton wrote that the term "power set" is rooted in the fact that card $\mathscr{P}A$ equals $2$ raised to the power card $A$.
As best as I can tell, the term Potenzbegriff (powerclass, later variation Potenzmenge, powerset) was introduced by Bernstein in the late 1890s (Cantor did not use it in his papers). In his Habilitation dissertation Untersuchungen aus der Mengenlehre (1901, published 1905) he states in the introduction (my translation):
"The introduction of the concept of powerclass, and the proof of the equivalence theorem to be mentioned later has now made it possible to arrive at conclusions that were previously only possible through laborious detours by an almost elementary calculation."
The word begriff literally means "concept", and intensional classes (falling under a concept) were not fully separated from extensional sets until Hausdorff's Grundzüge der Mengenlehre (1914), although Zermelo and others operated with the latter earlier. The "equivalence theorem" is now called the Schröder–Bernstein theorem, and Bernstein gives a proof in §1 of the dissertation. The original proof was given at Cantor's seminar in Halle in 1897, and does not survive (Dedekind gave a proof in 1887, but did not publish, Schröder's 1896 announcement was of a proof that apparently had a flaw). The set power (Potenz) is introduced in §2 as follows:
"If $M$ and $N$ are two sets, we call that set which - in the sense of a known expression - contains all combinations of elements from $M$ to the classes of $N$, the power $M^N$ ($M$ raised to $N$). Regarding the application to addition, multiplication and powerclasses of commutative and associative laws, they are the same as for finite numbers".
So the apparent motivation for Potenz (power) seems to be as commonly assumed, by analogy to raising numbers to powers. Bernstein does not use $2^N$ generally, or calls the class of all subsets the powerclass. But he does write $2^{\aleph_\alpha}$, meaning set cardinality (Mächtigkeit, often also translated as "power") in §9, when discussing the continuum hypothesis.
