Since this is a dystopian novel, I think it would be interesting to try to build a case for how Lowry shows that creating a perfect utopia is impossible through her use of various literary techniques. Here's an example of a thesis you could work with relating to this idea:
Through characterization and conflict, Lowry shows that the sacrifices needed to create a utopian society make a life of constant peace and stability impossible.
You could then talk about several issues within characterization. Jonas believes that his life is perfectly content and perfectly safe until he learns of the sacrifices needed to create this false reality. The old are celebrated just before they are euthanized, which no one addresses. He comes to love Gabriel, whom he learns will be released himself. He also learns that in the past, when memories were accessible to everyone, families loved each other, which is a concept that is lost in his society.
You could also examine Jonas's father, who seems to be an easygoing and kind person until Jonas realizes that he kills newborns who don't meet a certain standard—and he does so without the slightest sense of remorse. In various examples, characters portray a peaceful and stable sense of community, but horrendous sacrifices have been made in order to generate that sense of peace.
When examining the conflict that arises from trying to create a utopian society, you could look at the way carrying the memories alone has aged the Giver. He tells Jonas that he's not nearly as old as he looks but that the memories have aged him. You could also examine Rosemary's conflict; she has been so sheltered from pain, hunger, and poverty that when she is given those memories, she requests Release and effectively kills herself. Gabriel faces ongoing conflict because he isn't sleeping well through the night and isn't growing as quickly as he should be. Because of these perceived inadequacies, he is in danger of Release himself.
Characterization and conflict reveal that Jonas's Utopian society is a great disillusion. Great sacrifices are made, often in secret, to allow citizens to carry on with their ultimately meaningless lives, devoid of real emotions—positive or negative.
I hope this gives you at least a few ideas to consider for a literary analysis paper. Good luck!