Just checked The American Heritage Dictionary of Indo-European Roots (revised and edited by Calvert Watkins, Houghton-Mifflin, 1985). It has east, eastern, and Easter all derived from an Indo-European root aus-, "to shine."
This yields the Germanic root *aust- from which comes Old English east (with a long mark over the E), "east" (from "the direction of the sunrise") and the Germanic root *austr- in Old English easterne (with a long mark over the first E), "eastern." (The * indicates it is a reconstructed form, not found in any written text.)
The Germanic *austron- was a dawn-goddess whose holiday was celebrated at the vernal equinox, in Old English eastre (with a long mark over the first E), "Easter."
The same dictionary also has Latin aurora and Greek eos, both meaning "dawn," as deriving from the same Indo-European root aus-.
The reference to Eostre, if you dig is solely based on Bede’s off handed guess.
There is no evidence for such a goddess existence.
Think of it, thousands of written dedication to Tew, Woden etc but not a single one to any Eostre. And no contemporary source mentions her.onoy bede, 300 years after paganism has died out in England