What does Lily want her assignment to be in The Giver?
Lily says that she hopes that she will be assigned to be a Birthmother, because she thinks the Newchildren are so cute.
Lily is only a Seven, and she does not yet really understand what the different assignments involve. She will have to be an Eight before she even begins her volunteer hours, giving her an opportunity to experience something about the different possibilities first-hand. When Lily expresses the desire to become a Birthmother, her Mother is appalled. Mother sharply tells Lily, "Don't say that...there's very little honor in the assignment".
Lily has heard from a friend who volunteers at the Birthing Center that "Birthmothers get wonderful food, and they have very gentle exercise periods, and most of the time they just play games and amuse themselves while they're waiting" to give birth. To a Seven, this kind of life sounds kind of appealing. What Lily does not realize is that Birthmothers are allowed only three births, which take place within three years. After that, they spend the rest of their adult lives as Laborers, doing nothing but hard, physical work until they are old.
Birthmothers are essentially treated like animals, used for their reproductive capabilities for awhile, then for manual labor. Those who are chosen for this assignment generally have strong bodies but are not very intelligent. Mother most certainly does not want being a Birthmother to be her child's aspiration, and tells her why in no uncertain terms (Chapter 3).
Liily wants to be a Birthmother. She's in love with the idea of having babies. And as stated by the previous answer, she doesn't really have any idea of what it is really like for the Birthmothers after her 3 births are done. She may be thinking of a carefree existence, being treated well and eating what she wants, and not having to "work" too hard. But in reality, after her time as a Birthmother is done, the real work of being a common Laborer begins.
In the family that Jonas is in, being a Birthmother is seen as being beneath the status of their family. Not only would it be "common" but somehow a lowering of the entire family's status in the community for Lily to have a job with little honor.