Jonas is unable to transmit the memory to Lily.
Jonas’s rules tell him that he cannot discuss his training with anyone, and that of course means that he should not be transmitting memories either. However, he gets curious because Lily does not know what an elephant is. He wants to help her learn that the animals were not imaginary, because everyone believes that most animals like hippos and elephants were never real.
He placed one hand on each of their shoulders. With all of his being he tried to give each of them a piece of the memory …
But his father had continued to comb Lily's long hair, and Lily, impatient, had finally wiggled under her brother's touch. "Jonas," she said, "you're hurting me with your hand." (Ch. 13)
Transmitting the memory to Lily does not work because she does not have the Capacity to See Beyond. Unlike Gabriel, she is not literally related to Jonas. They are not related by blood (they do not both have the same light colored eyes, which is a clue). For this reason, she will never be able to see the memories like Jonas can.
Jonas is able to transmit a memory to Gabriel, however. He does this by accident. The baby is still fretting when he sleeps at night. They decide to put his crib in Jonas’s room.
He was not aware of giving the memory; but suddenly he realized that it was becoming dimmer, that it was sliding through his hand into the being of the newchild. Gabriel became quiet. Startled, Jonas pulled back what was left of the memory with a burst of will. (Ch. 14)
This is the first time Jonas transmits a memory to Gabriel, but it will not be the last. Gabriel does have the ability to receive memories, because Gabriel has the Capacity to See Beyond. Jonas and Gabriel begin to get very close, and Jonas sees Gabriel as his little brother. Later, when Jonas escapes with Gabriel, he will use the memories to calm him and sustain and protect them both.
The fact that Jonas cannot transmit a memory to anyone but Gabriel shows how few people have his special gift. Since we have very strong evidence that Jonas and Gabriel are related based on their light eye color, and the fact that The Giver has the same light eye color, they are also related to the one other person who has the gift, because she also has light eyes. Her name is Katharine and she is only six when Jonas leaves.
The unusual quality of the Capacity to See Beyond that Jonas has makes him able to receive and transmit memories. When he is unable to transmit the memory to Lily, it reinforces how different he is becoming from other members of the community. The distance between him and the others is growing. He has feelings and understandings that they will never have.