Oscar's life is largely based around searching. He is searching for clues to his Dad's life. He is searching for the answer to the last game he and his Dad played. He is also searching for the object his mysterious key belongs to. In all this searching he is, in a way, searching for himself. These things give him purpose and focus in his mind. His raison d'étre, so to speak.
Oskar found an envelope that encased a key, and on the envelope his dad had written, in red letters, the word BLACK. Being his curious self, he decided to search for the lock that the key opened, and the person with a last name of Black. He begins his search possibly finding the key and Black, or finding nothing.
In reference to the comment above, that whole scene then leads to another one, when Oskar goes to the art store to find out something about BLACK, and he discovers that it is common for people to write the color of the pen they're using. Because BLACK is written in red ink, this makes it seem more as if BLACK is a name, not a color. Oskar is looking at the notepad were all sorts of people write the color, or their name, on the pad in the color they're using, and he sees his father's name. He then treks through the whole store, finding his father's name everywhere. Searching. Searching for more about his father, his life, his interests, and his secrets.
"Every time I left our apartment to go searching for the lock, I became a little lighter, because I was getting closer to dad. But I also became a little heavier, because I was getting farther from Mom (52)".
Locks can be linked to the theme of searching, because in the process of searching for the mystery lock to the mystery key, Oskar truely is searching for himself, a connection with his father, and more details of his father's life and death.
While searching for the key, Oskar is really searching for signs of his fathers life. The proccess is long and greuling, but he learns a lot about his father that he would not otherwise know.
Searching brings people together. As in the case of Mr. A. R. Black (I didn't know where to find his first name, but I found his initials), Oskar unknowingly changes the man's life by bringing hearing back to him and inviting him to come along on his journey. However, searching also tears apart, as is the case with Oskar's mother being more and more shut out from Oskar's life, even to the point that his search becomes his ultimate reason for living, making it higher in his list of priorities than his mother, grandmother, and, ultimately, his father. He's so obsessed with this search that he won't even live for anything else, despite how much more significance they bring to his life.
"He told me to go up to the observation deck of the empire state building, and as he walked around New York, he'd occasionally shine the light up at me so I could see where he was." (252) This quote is from Ruth Black, talking about what her husband used to do for her. When he died, and was no longer able to shine the searchlight through the streets of New York, she moved into the Empire State Building. So, maybe she lives there to continually search for her dead husband, or to hold onto his memory longer?
It becomes apparent, especially near the end of the book, that the ultimate reason for the search was to simply have a search. Because Oskar never truly finds anything to help with the issue of his father that relates to the key, which was the original object of the search, it is clear that the search itself was what truly gave him the peace to accept his father's death. When Oskar speaks to his grandfather for the first time since he has discovered what the key unlocked, he says "'I wish I hadn't found it.' 'It wasn't what you were looking for?' 'That's not it.' 'Then what?' 'I found it and now I can't look for it.'" This short conversation shows how the ends didn't justify the means in Oskar's case because the search revealed more for Oskar that the object of the search.