The hare had several symbolic elements, a common one was that the hare represented virginity (there was a misconception that hares/rabbits could produce offspring without sex) and so the hare can be seen included in images of the Virgin Mary. The hare was also a symbol of our salvation, Proverb 30:26 says "the badgers are a people without power, yet they make their homes in the rocks;", the Hebrew word shafan translated as "badgers" in the NRSV probably refers to the hyrax, St. Jerome translated it into Latin as lepusculus, "the hare" or "the rabbit"; likewise Psalm 104:18 says much the same about the hyrax/rabbit/hare/shafan finding safety on top of the rock.
The following is from St. Augustine's exposition on Psalm 104 (well, 103 in the Latin),
"What of the hare? What of the hedgehog? The hare is a small and weak animal: the hedgehog is also prickly: the one is a timid animal, the other is covered with prickles. What do the prickles signify, except sinners? He who sins daily, although not great sins, is covered over with the smallest prickles. In his timidity he is a hare: in his being covered with the minutest sins, he is a hedgehog: and he cannot hold those lofty and perfect commandments. For "the loftiest hills are for the stags." What then? Do these perish? No. For so "is the rock the refuge for the hedgehogs and the hares." For the Lord is a refuge for the poor. Place that rock upon the land, it is a refuge for hedgehogs, and for hares: place it on the sea, it is the home of the coot. Everywhere the rock is useful. Even in the hills it is useful: for the hills without the rock's foundation would fall into the deep...."
Taking the rock as symbolic of Christ our rock and refuge, and the hare as symbolic of us in our weakness and timidity, Augustine sees here our salvation in Christ.
The hare becomes, then, a symbol of ourselves, specifically of our salvation upon Christ our rock.
How the hare, a symbol of Mary's virginity or of our refuge in Christ, came to become the Osterhase (Easter hare) in Germany is a good question; though the game of having women and children search for eggs which symbolized the women going to the empty tomb dates at least to the 16th century (Luther was particularly fond of this so I've read). The idea of the Osterhase dates to, I think, about the 17th century.
It's safe to say that whatever pagan connotations the hare had in pre-Christian Europe didn't survive, and the hare's place in Christian iconography and symbolism was already well established before there was an Osterhase at all.
It also remains a complete fiction that the Anglo-Saxon Eostre was associated with hares and/or eggs--there's simply no evidence of this in the material we have (Bede), and is just a bit of 19th century speculation which continues to brandied about as fact.
-CryptoLutheran