The name Esther - אסתר - is connected to the Babylonian deity Ishtar (yes, the same name as the notoriously unsuccessful movie.) They both derive from the Indo-European root ster, and the related Semitic root ʿṯtr which gave us the Greek goddess Aphrodite and the Phoenician goddess Astarte עשתרת. That same root gives us the English words star, astral, stellar and disaster (not in the stars.)
I briefly mentioned Aphrodite, but didn't focus any further on that name. The Online Etymology Dictionary provides this origin:
Greek goddess of love and beauty, personification of female grace, 1650s; the ancients derived her name from Greek aphros "foam," from the story of her birth, but the word is perhaps rather from Phoenician Ashtaroth (Assyrian Ishtar). Beekes writes, "As the goddess seems to be of oriental origin ..., the name probably comes from the East too. .... It may have entered Greek via another language." He concludes, "[I]t seems possible that the name came from the one languages [sic] which on historical grounds we should expect to be relevant: Cypriot Phoenician."
Klein agrees that the idea that the name derives from aphros ("foam") is a folk etymology, but does suggest that perhaps her association with foam caused the change in pronunciation from Ashtoreth to Aphrodite. He gives other examples of "sh" turning into "f." He points out that garlic in Hebrew is שום shum, but in Arabic it is either thum or fum. Similarly, the Russian name Feodor derives from the Greek Theodore.
From Aphrodite, according to some theories, we get the name of the month of April. Klein writes that April, in Latin Aprilis, comes from Greek Ap(h)ro, a short form of Aphrodite, and so was "the month of Aphrodite." The Online Etymology Dictionary suggests (among other possibilities), an Etruscan origin, but still coming from Aphrodite.
So we've shown connections between Esther and April, but one word I was surprised to discover isn't related is Easter, which usually falls in this time period as well. However, here is Easter's etymology:
Old English Easterdæg, from Eastre (Northumbrian Eostre), from Proto-Germanic *austron-, "dawn," also the name of a goddess of fertility and spring, perhaps originally of sunrise, whose feast was celebrated at the spring equinox, from *aust- "east, toward the sunrise" (compare east), from PIE root *aus- (1) "to shine," especially of the dawn.
I guess that connection wasn't in the stars...