Our main source for the word 'Easter' is Bede, writing around 730AD in On the Reckoning of Time who says:
In olden times the English peoplefor it did not seem fitting that I should speak of other nations' observance of the year and yet be silent about my own nation'scalculated their months according to the course of the Moon. Hence, after the manner of the Greeks and the Romans, [the months] take their name from the Moon, for the Moon is called mona and the month monath. The first month, which the Latins call January, is Giuli; February is called Solmonath; March Hrethmonath; April, Eosturmonath
Eosturmonath has a name which is now translated "Paschal month" and which was once called after a goddess of theirs named Eostre, in whose honour feasts were celebrated in that month. Now they designate that Paschal season by her name, calling the joys of the new rite by the time-honoured name of the old observance.
The problem is the existence of the said goddess, who outside of Bede doesn't really exist. Bede has his own historical problems, and recognizes in this instance that he's not really sure about it and it's Bede's interpretation, rather than being supported from outside evidence.
The other linguistic issue, is that Anglo-Saxons named their months agriculturally, not according to deities (see the list here[1] ). So it's rather strange that Eostre should suddenly pop up and be a month name. You also see the problem with Hredmonath - outside of Bede, there's no knowledge of the goddess (if that is indeed what it's referring to). Philip Shaw recently published a monograph on this[2] and attempts to link 'Easter' with the Old Norse 'austr' the "East". The problem is that it's one thing to show a linguistic link - it's another to show that there actually was something called Eostre that was worshipped.
Yet another issue is that Charlemagne is responsible for renaming all the months of the year, and
April is turned into 'Ostarmanoth' - considering that Charlemagne has been waging war on the Saxons[3] and converting them somewhat by force, it's a bit strange that he'd suddenly name the month after a pagan goddess.
So it makes much more sense that Eostermonath to be translated as 'the month of opening' - which fits with the time period (spring), rather than the goddess.