Apparently first introduced by Weierstrass in [Winter 1862/63 lectures](http://archive.org/stream/encyklomath202encyrich#page/n272/mode/1up) published by H. A. Schwarz (1881, [1885](https://archive.org/stream/acq9098.0001.001.umich.edu#page/10), [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. (...)

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](http://bibliothek.bbaw.de/bibliothek-digital/digitalequellen/schriften/anzeige/index_html?band=10-sitz/1882-1&seite:int=465)). Attribution in e.g. (Schwarz student) H. Hancock's book *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
$$
\wp(u)\qquad\text{or more simply}\qquad\wp u.
$$