The letter $\wp$; Name & origin?

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. (...)

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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).

The Weierstrass p has a strong similarity to p in the Sütterlin alphabet, which had been developed in Prussia and, as Weierstrass worked in Berlin (the capital of Prussia), it may well be that that is the origin of the letter.

Books were printed in Fraktur, where the p looks quite normal, i.e., quite different from a handwritten Sütterlin p which could explain, why it hasn't been replaced in the publication of Amandus Schwarz.

But I am not a historian, so this answer is a bit speculative, albeit reasonable.

The Sütterlin alphabeth was introduced by the graphic designer Sütterlin in the 1920s and was the handwriting in German schools 1935-1941. It was a slightly simplified version of Kurrent. Before that time the usual handwriting style in Germany (since the 1500s) in german language was Kurrent (also known as Kurrentschrift, Alte Deutsche Schrift ("old German script") and German cursive) which was used also in Switzerland till 1900s and in Austria till 1941 (and still taught in schools after 1945; I learned it besides the Ausgangsschrift in elementary school 1955-1959).

The letter $\wp$ is, originally (in my eyes, and as I learned in lecture classes), just a capital P in Kurrent, written in a slightly elaborate way. I would say that the Weierstrass-P symbol $\wp$ was introduced by Donald Knuth as a special math TeX-symbol, :) . I would have used $\mathfrak P$ before learning about $\wp$ in the question.

See here for six letters by Gauss written in Kurrent.