The name refers to the concept of an anharmonic ratio, or cross-ratio. Four points $A,B,C,D$ are called equianharmonic if their cross-ratio is a cube root of 1. In that case the 6 cross-ratios obtained by combining the four points in all possible ways are the same ("equal anharmonic ratio" = "equianharmonic").
The curve $y^2=x^3-1=(x-p_1)(x-p_2)(x-p_3)$ has $p_1=1$, $p_2=e^{i2\pi/3}$, $p_3=e^{i4\pi/3}$ equal to the three cube roots of 1. The four intersection points of a line with the four tangents, drawn from any point on the curve, are equianharmonic.
If you read German, you might want to know how Wiener described this when he introduced the concept in 1901:
