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 *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
$$
\wp(u)\qquad\text{or more simply}\qquad\wp u
$$

or R. Godement's *Analysis I* ([2004](https://books.google.com/books?id=yqkSBwAAQBAJ&pg=PA181&dq=%22the+famous+function%22)), 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 inventor<sup>65</sup> and retained by posterity. (...)

>----------
><sup>65</sup> 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](https://mathshistory.st-andrews.ac.uk/Biographies/Pincherle/) in Berlin: ([1899-1900, Chap. XXII](https://archive.org/stream/lezionisullateor00pincuoft#page/14)).