How can one organically develop a strong relationship (father-son, to be precise) between 2 strangers over the course of a novel without making it seem contrived?
Have them face the worst possible moment of their lives together.
For example, if they were in an earthquake and got trapped in the basement of a building, just the two of them, with no way to call for help and limited food and water and the air running out and both are injured in some way … if they get out of that together, they will be each other’s closest friend.
Relationships are contrived. They are contrived by the people in them. They are formed because one of the parties sets out, more or less deliberately, with more or less forethought, to create the relationship. In other words, relationships are the result of courtships.
We are social creatures and we court other people all the time. In many cases, the courtship is mutual, or becomes mutual.
If you attempt to have the relationship form without one character courting the other, that will seem contrived. It will seem contrived because the relationship will appear without either of the characters doing the things that form relationships. Figure out who wants the relationship and why and have them act accordingly to try to create the relationship.
In other words, have one character court the other for believable reasons.
There should be a natural draw between them, as if they would normally be friends. If the younger character has a backstory similar to the older character or someone the older character cares/cared about, that would help.
The "father" role is very tough to build with a stranger, but mentor or "uncle" is easier.
Keep in mind bonds are formed through shared experiences, so the relationship should build over time as the younger character doesn't listen and has to be rescued, or his ignoring the older character gets people he cares about hurt.
This depends a lot on the context of course (not to mention the genre, if applicable), but try to see it from as a realistic perspective as possible - if indeed your goal is to make it appear natural.
How do people get closer in real-life?
Sociological causes: When they have to be united against something else. There is an element of tribalism involved, but for a father-son relationship this should not appear as too peculiar. Example: Father and son have to unite against a corporation threatening to confiscate their land
Personal causes: There might be a specific reason (that the readers and/or characters might only partially know of) for which one character is attracted to the other and wants to be around them. Example: Father has a terminal disease but he does not want his son to know about, so that the son can feel connection, not pity. There's a lot in such scenarios you can use dramatically, to enhance tension. Particularly if you allow your audience to know about it (while the character does not), there are strong elements of tragic irony at play
Miscellaneous/Forced causes: Sometimes people can be forced to be together, for a variety of reasons. Think of two people who are castaways together (somewhat of a cliche, I know, but only to give you an idea of how it works). To develop this organically, this should ideally evolve into more personal reasons. But it can be a start.
The essence of a good father-son relationship is sacrifice by the father on behalf of his son, it must be a truly altruistic sacrifice, with no reward expected.
The essence of a bad father-son relationship is the opposite; the father is selfish and demands (or forces) sacrifice from his son to meet his own selfish wants and needs; so he will force his choice of everything on his son: Controlling his life, his leisure and interests, his friends, his romances, his job choices, everything. Often to make his son a clone of himself, or something that reflects on him: My son WILL be a doctor, or a lawyer, or a CEO, like it or not. Or WILL be a criminal and member of my criminal enterprise. This is the way some sons grow to hate their father and everything about him.
To forge a good father-son bond with a stranger, you need to emulate this. We'll call the father figure 'father*'. The father* needs to do something altruistic for the son, preferably repeatedly. The son needs to feel gratitude for that; and this will be easier if the son's real father was of the selfish, self-centered type, and the son really doesn't like his real father.
The altruistic act doesn't have to be momentous, but obviously should start the relationship. For example, the father* may pick up the son hitching on the road.
Son: "How far you headed?"
Father*: "As far as Stinson, have to pick something up."
Son: "Alright, Stinson's thirty miles closer than I am now." He throws his backpack into the bed of the truck, and climbs in the passenger seat.
Father* gets back on the road. "Where are you headed?"
"Albertville, but twenty miles at a time, if that's what it takes."
Father* nods. Conversations. They stop in Stinson. Kid's name is Mark. Father* is Walter.
Mark: "End of the line. Well, thanks Walter, that's a help."
Walter: "If you want to hang loose for twenty, I'll take you on to Albertville."
Walter: "Yeah, really. Get in, We'll go get my parts. Anyplace good to eat in Albertville?"
Think of it like the road trip that starts "When Harry met Sally". In that case a time for future-lovers to get to know each other, but it starts a lifelong friendship. But in this case, Walter is doing the kid a favor. In their conversations it is important to see conflict between them, but like Harry and Sally, it is friendly philosophical disagreement. Don't make Walter an infallible fount of wisdom; make both Walter and Mark learn something. Mutual benefit is the foundation of friendship (you like each other's humor, and share likes and dislikes and some general philosophy, but are still different, not boringly exactly the same), and you need to make strangers friends before they can love each other; which is what you want for a good father-son relationship.
What leads to LOVE is shared emotional events, and for a parent-child relationship, more selfless acts on the part of the parent. Walter needs to feel invested in Mark's success. One way to do that is to make Walter a part of Mark's success, as an older male he has more contacts and more influence and more knowledge about how the world works, which is going to waste until Mark comes along; but can be freely given by Walter and benefit Mark enormously.
That is all in the metaphor of the ride to Albertville: Walter gives up his time and pays for the gas, and maybe lunch on top of that in some joint they both like (give them something in common) and doesn't demand anything from Mark in return, he enjoys his company. But of course he gets something from Mark in return, some gratitude for the gift. Turns out, Mark's stay in Albertville is temporary, and this is the beginning of a relationship. But don't make Mark a parasitic kid, either, he needs to be a competent person on his own. The type of son a father would be proud of.
The characters should start the story with a need --probably subconscious --for someone in the opposite role. The son-figure may not think he needs a father, but he's in desperate of advice, guidance, a male role model. The father figure may have been running from the commitment a family represents, but he's keenly feeling how empty his life is. If there's no void in their lives, it can't be filled.
I think that you need to define your two character's wants, needs, fears, and frailties clearly. Then, have them interact according to their natures, with appropriate levels of openness and vulnerability given their situation. The more circumstances out of their control drive their interaction, the less genuine their relationship will feel. But, if they are in a voluntary situation and are choosing to interact with one another, then as they face challenges and vicissitudes together, their bond will grow because it naturally does between humans in those situations. As long as you know their natures and you let the act and react honestly, you will find that they form as tight and genuine relationship as you can imagine.
The child character needs to have a need for a father figure.
People have many friends, but generally few parents. Children in fantasy books often have no father figure at all. What is the child character missing that they need a father figure? Do they need protection? Validation? Someone to explain how the world works?
The parent character needs to have a reason to believe intervening in the child's life is acceptable.
People don't take in random young kids at the playground. That's called kidnapping. The parent character needs to have a reason to think taking on a parental role is justified. Maybe the child is clearly wandering around a dangerous place alone. Maybe the child asks for help directly.
The parent character's involvement will generally start out small and then escalate. If they meet in a dangerous place, the first goal is to get the child to a safe place. If there is no obvious safe place, the parent tries to find one. If that fails, they're stuck with the kid. If the child instead needs validation, the parent character may start out as a teacher, performing an assigned job. Slowly (perhaps through the child asking for help on personal projects that have a more sinister context, perhaps through running into the child in dangerous places outside of teaching) the parent character comes to realize that the child needs help outside of the professional setting.
Friendly relationships usually develop in stages, from strangers to acquaintances to friends to an essential part of life. In a high-pressure environment the stages may be relatively short time-wise, but the important thing is each stage seems perfectly reasonable given what has come this far. The parent has already saved the kid from a monster, or accidentally gotten him in trouble, it would be unethical to just leave the child here now, for example.
Make the older guy more forgiving. Make the older guy more forgiving. Long story short, the older man doesn't want the kid to drive fast. But he is in charm, drives fast, and got into some huge trouble with some rich guy. But instead of beating him up, he tries to ease his pain, etc., you can do whatever you like, from sci-fi to horror or old time.