When we look at lists with opening lines, they all seem to be very different from each other and have little or nothing in common. Some can be grouped together into categories, such as stating an eternal principle ("Happy families are all alike; every unhappy family is unhappy in its own way."), but others clearly fall into other categories, and these categories do not overlap ("It was a queer, sultry summer, the summer they electrocuted the Rosenbergs, and I didn’t know what I was doing in New York.").
But opening lines only appear to be different because we make the mistake of looking at them out of the context of the novels that they are a part of. Once we understand that all beginnings of all novels are hyotheses that the novels then set out to test, we understand the similarity between the first sentences.
Leo Tolstoy presents the hypothesis that all unhappy families are different in how they are unhappy, and then proceeds to narrate the stories of three families to show us that his hypothesis is true. Sylvia Plath presents the hypothesis that her derealisation and mental problems are unrelated to the society and politics of her time, and then proceeds to show us that this hypothesis is false.
So if you want to derive your opening sentence from your story, all you have to do is understand what the hypothesis is that your story serves to prove or disprove. All novels are constructed – consciously or unconsciously – on some fundamental assumption or basic premise. Your first sentence must state this premise.
In a comment to this answer @ChrisSunami has remarked that when he tries to apply my idea (that the first sentence is a hypothesis that the book tests) to actual books, he finds it difficult to find any that actually fit the pattern. I have asked him to provide some examples, and he has suggested five, from which I will discuss the three that I am familiar with.
Michael Ende's novel Die Unendliche Geschichte [The Neverending Story] begins:
This has been translated into English as:
(Literally, the German shop sign reads: "Antiquarian Book Seller, Owner: Karl Konrad Koreander".)
This opening is special, in that it is not simply text, but an image of text printed on glass and seen from the other side. Now "the other side" is not an innocent phrase when it comes to literature. You immediately think of the celtic otherworld, the mythic underworld, of the hero's journey through death and rebirth, and of classic fiction such as Lewis Carroll's Through the Looking-Glass. What the Neverending Story begins with is the motif of the mirror, and more specifically the motif of looking out at the "real world" from the mirror's other side.
Michael Ende has spoken extensively of mirrors and mirror effects in interviews and elsewhere, and even dedicated a book to them, Der Spiegel im Spiegel [The mirror in the mirror], a collection of surrealistic narratives, which, as he has stated, deals with the question: "What is reflected in a mirror which is reflected in a mirror?" The answer that Ende gives is: An infinite play of reflections which reflect each other: a neverending story.
The Neverending Story itself is such a mirror reflected in a mirror. The narrative claims that the book the reader is reading is in fact the book that the protagonist, Bastian, is shown to be reading after he finds it in the bookstore whose window we see at the beginning of the text. Bastian later finds himself to be a character in the book he is reading, adding another, deeper level, and the reader Bastian can influence what the characters in the book that he reads do.
Michael Ende wrote (in a note found in his estate) that "when two readers read the same story, they do not read the same. Each of them brings himself into the reading, his thoughts and associations, his experiences, his imagination, his intellect. You might say, the book is a mirror, in which the reader sees his own reflection."
And that is the hypothesis: That a book is a mirror in which the reader sees his own reflection, and in The Neverending Story Ende sets out test whether this idea might be true and to what extent.
Samuel R. Delany's Dhalgren is a novel about reading a book that tells your own story as well. The protagonist, Kid(d), finds a notebook that opens with the same words as Dhalgren, but the story Kid(d) reads diverges from the story that we are reading (and he experiences), so Kid(d) begins to revise the text, until his revision echoes the end of Dhalgren verbatim.
Delany has stated that he suffers from delusions and spend time in a mental institution. As Wikipedia says, Delany "has repeatedly spoken and written of seeing burned-out sections of great American cities that most people didn't see, or even know exist", and that "Dhalgren is a literary exposition of all these experiences for the 'normal' reader."
Dhalgrens first sentence – "to wound the autumnal city." –, obscure and hermetic as the beginning lines are, clearly shows the reader (even if only in hindsight, when he comes to the end), the circular structure of the tale. It names the hallucinations that its author suffered from (the "burned-out sections of great American cities") and, together with the novel's end ("I have come to"), this sentence ("I have come to wound the autumnal city.") states that by writing about how he experiences reality the author, Delany/Kid(d), rewrites reality for his readers as well.
Dhalgren is a novel about writing, and its hypothesis is that life is something alike to rewriting: you take the story that is already there (the world that you are born into) and rewrite it to make it your own. Or, in the words of the novel's first sentence: You wound the world.
Vladimir Nabokov's Lolita, which begins with "Lolita, light of my life, fire of my loins.", hypothesizes – by identifying Lolita with life and sex: Lolita = life = sex – that whatever energy sustains life is given to a man by a woman through sex. If a woman doesn't gift sex to a man – or if the object of his desire is legally unavailable, as in Humbert's obsession with an underage girl –, he dies. And the novels sets out to show just that.
Nabokov has explored the same topic of hebephilia leading to a man's demise in many of his works, including his pseudo-autobiography Look at the Harelquins!. The earlier ones make the theme very clear:
In 1928 Nabokov wrote a poem named Lilith (Лилит), depicting a sexually attractive underage girl who seduces the male protagonist just to leave him humiliated in public. In 1939 he wrote a novella, Volshebnik (Волшебник), ... it takes place in Central Europe, and the protagonist is unable to consummate his passion with his stepdaughter, leading to his suicide." (from Wikipedia).
The topic in Lolita is child sexual abuse, but the hypothesis outlined in the first sentence is that if the obsessed man is not able to consummate his desire, he dies (or lives on broken).
Let me note that I didn't claim that the hypothesis is necessarily plainly stated. When you consider my answer, please keep in mind that my question was, "How to derive a first sentence from a novel." My answer to my question is that to derive a first sentence from a novel you could – and I actually believe that you should – find the hypothesis that your book is trying to answer. That is, the hypothesis is something that the author is (or should be) aware of and may, if he is so inclined, use when he wants to derive a first sentence from his story. I don't say that the hypothesis is something that a reader will always easily understand, and in fact I believe what hooks a reader is that he does not understand, but feels there is more to that sentence that what it says on the surface.
You all, who object that you cannot find this principle in the books that you have read, lose sight of what this site and this question are about. The purpose of my answer is not that you now go out and attempt to prove me wrong by finding books where this principle doesn't hold. We are not Literature.SE! What you should be doing is prove me wrong by applying it to your own writing. Take the book you are working on, see what hypothesis you are trying to test, and write a sentence that exemplifies this theme. You'll have an intriguing opening sentence that fits your book.