This species was originally called Dendroclava dohrnii in honor of Anton Dohrn (as Rafael noted in his comment). This is affirmed by August Weismann in his book:
Diesen Namen schlage ich zu Ehren des Gründers der vortefflichen
zoologischen Station in Neapel für eine bisher unbekannte Clavide vor,
welche im Golf von Neapel in grösserer Tief lebi. (Die Entstehung
Der Sexualzellen Bei Den Hydromedusen, pg. 26)
I propose this name in honor of the founder of the excellent
zoological station in Naples for a hitherto unknown clavid, which
lives in greater depth in the Gulf of Naples.
The parent of Turritopsis dohrnii is Turritopsis mccrady, and the reason given for assigning this name is as follows:
The present sub-genus, for which I propose the name Turritopsis, is
distinguished from Turris by the following peculiarities: The swimbell
is remarkably thin, having no accumulation of transparent tissue above
as in Oceania and Turris, but the whole appears to be a muscular mass
covered by epithelium within and without. (pg. 56)
As far as the name Turris, that is attributed to René-Primevère Lesson. Although Lesson doesn't give his reason for naming it as such, I agree with Job Rozemond that it probably means tower. The description given by Lesson kind of sounds like a tower:
Ombrelle allongé, cylindrique et pyramidal, tronqué à la base que a
huit tentacules, dilatés et échancrés à leur attache; muni de quatre
lignes vasculaires, longitudinales. (Zoophytes, pg. 283)
Long umbrella, cylindrical and pyramidal, truncated at the base with
eight tentacles, dilated and indented at their attachment; provided
with four longitudinal vascular lines.