In Weierstrass notation, the principal elliptic function $\wp$ is a solution of the differential equation $$ (\wp')^2= 4\wp^3 -g_2\wp -g_3. $$ The case when $g_3=0$ is called lemniscatic (it corresponds to a square lattice), and the case $g_2=0$ is called equianharmonic (it corresponds to a hexagonal lattice). The origin of the first term is clear: it comes from the problem on the length of the Bernoulli's lemniscate.
What is the origin of the term "equianharmonic"?
The term "equianharmonic" seems out of date: checking Google ngram shows a strange pattern: its usage experienced a peak around 1860 and was much more frequent than "lemniscatic" until 1980s, and nowadays it is used about 10 times less frequently than "lemniscatic". Also the sizes and details of Wikipedia articles confirm this.