Apparently first introduced by Weierstrass in Winter 1862/63 lectures published by H. A. Schwarz (1881, 1885, 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. (...)
The letter and a reference to Schwarz's notes also appear on the first page of Weierstrass's paper Zur Theorie der elliptischen Functionen (1882). Attribution in e.g. (Schwarz student) H. Hancock's book 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 $$ \wp(u)\qquad\text{or more simply}\qquad\wp u. $$