Apparently first introduced by Weierstrass in lectures published by H. A. Schwarz (1892, 1893, §9):
Mit der Sigma-Function $\mathfrak Su$ ist die Pe-Function $\wp u=\wp(u\mid\omega,\omega')=\wp(u;g_2,g_3)$ durch die Gleichung $$ \wp u=-\frac{d^2}{du^2}\log\mathfrak S u=\frac{(\mathfrak S'u)^2-\mathfrak S u\mathfrak S''u}{\mathfrak S^2u} $$ verbunden. (...)
Attributed in e.g. H. Hancock (Berlin Ph. D. with Schwarz), Lectures on the Theory of Elliptic Functions (1910), p. 309:
(...) the function which we thus have was called by Weierstrass the Pe-function and denoted by $$ \wp(u)\qquad\text{or more simply}\qquad\wp u. $$