Finding a lost item of great importance, such as one's wedding ring or a child's safety blanket
Finding a historical or mythological artifact (the Holy Grail, Excalibur, the fountain of youth, etc.)
Tracking down an abducted loved one
Finding the child one gave up for adoption
Finding the family one was separated from as a child (because of civil unrest, a natural disaster, being kidnapped, etc.)
Locating a lost civilization
Reviving an old truth that has been forgotten or rewritten
Reconnecting with one's roots that have been forgotten, abandoned, or one never knew about

There are five basic human needs that, when missing from a character's life, could motivate them to pursue this goal. The following needs are all possibilities, but only one of them should be the primary driver for any given character. For more information on the relationship between human needs and outer motivations, please see this Character Motivation tutorial.

Self-Actualization: When finding the lost thing is a matter of justice or "rightness" for the character (such as a search for truth or vindication), their need to be fully actualized may drive them to pursue this goal.
Esteem and Recognition: A character can be driven to find something lost if doing so will bring them great esteem in the eyes of others.
Love and Belonging: This need could drive a character to find the people, cultures, or traditions they've lost.