I think this thesis almost demands a comparison between meaningless and meaning as portrayed by the book. The previous posts give good suggestions as long as you are tying each character or event (example) to meaningless or meaningful.
You could also look for some key quotes that perhaps show the difference between Montag's meaningless life (at the beginning of the story) and his search for something more (toward the middle) and then something from the end of the story that gives evidence of "meaning."
Do you know why books such as this are so important? Because they have quality. And what does quality mean? To me it means texture. This book has pores. (Faber, The Sieve and the Sand)
I like the previous thoughts. I would add that examining Mildred's role in the novel might help enhance the analysis of the theme a bit. Mildred lives a life that lacks critical thinking. Her conformity with social expectations and governmental desire is evident and she lives a life that is vacuous, at best. She is content with her life of serials and being part of the herd. When she begins to feel threatened with her husband's own engagement in critical thinking, she doesn't understand why she feels the way she does. Her lack of critical thinking abilities compels her to take an overdose of sleeping pills. This indicates that the true pain of a life devoid of critical thinking is that one feels that there is a problem, but they lack the vocabulary and articulation to fully comprehend it. This seems to be a worse fate than one who is a critical thinker and persecuted for it.
In Fahrenheit 451, books are banned. There isn't a clear explanation of why but Bradbury toys with this idea to play on this theme. He questions why there are two kinds of people, those who are uninterested in reading (critical thinking) and those who are hostile against books because they feel inferior to a society that is more well read. This is where knowledge vs ignorance comes into play. If you promote ignorance or a lack of critical thinking/ goals, you will create a population that is equal yet there will never be greatness. Everything will be the same. Therefore, Bradbury battles this idea of a society and states that without a goal in your life, it will ultimately be meaningless. Make meaning with your life, think, read and you shall live in a full society.