Why are Boolean Algebras called "Algebras"?

Solution 1:

Because Boole himself introduced the word "algebra" into the subject.

The term "algebra of logic" appears in Boole's 1854 book on Laws of Thought:

Let us conceive, then, of an Algebra in which the symbols x, y, z, etc. admit indifferently of the values 0 and 1, and of these values alone. The laws, the axioms, and the processes, of such an Algebra will be identical in their whole extent with the laws, the axioms, and the processes of an Algebra of Logic. Difference of interpretation will alone divide them. Upon this principle the method of the following work is established.

Boole strongly emphasized the relation between logic and algebra. References to algebra and its correspondence with logic permeate the book.

Other writers continued to use "algebra of logic" for Boole's system and its later simplification to what is now called Boolean algebra. For example, MacFarlane Principles of the Algebra of Logic (1874), C.S. Pierce "On the Algebra of Logic" (1880), and E. Schroeder Algebra der Logik (1890).

In addition to the analogy that Boole had observed with ordinary algebra, there is an equivalence of Boolean algebras with rings satisfying $x^2=x$ for all $x$, which are equivalent to some algebras (in the modern sense) over the 2-element field.

Solution 2:

They are algebras, in the sense of universal algebra (where "algebra" is basically synonymous with "first-order structure", except that it requires the language to have no relation symbols).

In fact, I think this notion of algebra = algebraic structure long preceded the definition of algebra as (roughly) a module with multiplication. And it was in this context that Boolean algebras were so named - that is, just because of their "algebraic" nature.