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 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 $$ \wp(u)\qquad\text{or more simply}\qquad\wp u $$
or R. Godement's Analysis I (2004), p. 181:
(...) the famous function $$ \wp(u)=1/u^2+\sum_{\omega\ne0}\left[1/(u-\omega)^2-1/\omega^2\right] $$ of Weierstrass (it already appeared in Eisenstein), with a $p$ which smacks of the gothic, of the italic and of the cursive, chosen by the inventor65 and retained by posterity. (...)
65 His biography in the DSB tells us that in the course of his fourteen years of high-school teaching he had to teach mathematics, physics, German, botany, geography, history, gymnastics “and even calligraphy”.
Note added: While I don’t know of a handwritten specimen by Weierstrass himself (asked about in comments by @NateEldredge and @ManfredWeis), there are a few in lecture notes of S. Pincherle who had studied with Weierstrass in Berlin: (1899-1900, Chap. XXII).