This one something of a mystery. The name Eshima is almost certainly derived from the much more well-known Enoshima Jinja: the possessive particle "no" has been dropped (not at all unusual) and an older form of the kanji for "shima" (island) is used. No information is given in the shrine grounds and a Google search in Japanese turned up just 35 results. Two sites account for five of the mentions and the others are all irrelevant. To be fair, a look at the photos below may well explain the lack of interest, visually appealing this shrine is not.
The name of the Enshrined Deity, Ichikishima-hime-no-mikoto, is taken from one of the two relevant results from the Google search (Japanese), but it is described there as "presumed" and there is no further explanation, although she is the main goddess at one of three main shrines at the Enoshima Jinja. Note that Benzaiten has also been equated with Ichikishima-hime-no-mikoto. Further backing for this association is provided by the fact that the enshrined goddess in the hokora is playing a lute, which is the way the goddess Benten is usually portrayed, and Enoshima Jinja is known as one of the three major centres of Benten worship in Japan. The name Mizuko Yamazaki is engraved on the goddess' plinth: given that the shrine is in the Mizuko district this name is probably of local rather than national significance.
Eatliest record of: Unclear
Annual Festival: ?