Apparently first introduced by Weierstrass in lectures published by H. A. Schwarz ([1892](https://archive.org/stream/fomundezumderfun00weierich#page/10), [1893](https://archive.org/stream/formelnundlehr01weierich#page/n21), §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](https://archive.org/stream/lecturestheorell00hancrich#page/n340)), p. 309:

>(...) the function which we thus have was called by Weierstrass the *Pe-function* and denoted by
$$
\wp(u)\qquad\text{or more simply}\qquad\wp u.
$$