Lina and Doon's friendship in The City of Ember grows for one reason. They are ultimately searching for the same thing. What are they both searching for?
Unlike many of their coresidents in Ember, Lina and Doon are curious and willing to attempt to investigate reasons and possible solutions for the increasingly frequent and long losses of electricity. Because this goes against the prescribed patterns of behavior for citizens, they face numerous challenges and repercussions at various levels of severity.
Ultilmately, both Lina and Doon are searching for self-fulfillment and the freedom to express their individual personalities. They both are concerned for the future of Ember and, in their own ways and in cooperation, act upon those concerns. They share personal values of honesty, determination, resourcefulness, and a lack of fear in the face of the unknown. In Ember, these traits are not considered as assets, and indeed are discouraged. Lina and Doon are unwilling to suppress their abilities to conform to the expectations of their society, particularly as they begin to recognize that they have found the means of saving Ember from eternal darkness.
Lina and Doon are both unsatisfied with the job assigned to them. That is the start of their friendship. They are searching for answers to their dysfunctional society. They want to know the truth about their community's past. But even more than that, they are searching for freedom from their society's restraining rules.