Why does Homer's father want him to work in the mine in "October Sky"?

Coal mining towns are small and, historically, desperately poor. Yet, mining coal is what the town has, and it holds to it as tightly as possible. It provides sustenance for those who work in the shafts and for those who provide local services to the mine. For many families, especially multigenerational families for whom mining coal has been all that they have known, considerations of other professions are kept to a minimum. That was true throughout the nineteenth and much of the twentieth centuries. In Homer Hickam’s memoir Rocket Boys and in the film adapted from that book, October Sky, the Coalwood, West Virginia, is the quintessential coal town, with economic and much social life centered on the local coal mine. Children of coal miners were expected to follow their fathers’ paths, finishing high school and then working in the mines.

October Sky includes numerous examples of dialogue in which the expectation that young Homer will follow his father’s footsteps into the local mine despite the boy’s fascination with rocketry and dreams of a life outside of Coalwood. Just as numerous are Homer’s protestations against following his father, John’s path.

Coal mining may be your life, but it's not mine. I'm never going down there again. I wanna go into space.

John’s attitude towards his son’s interest in rocketry slowly—very slowly—evolves as the film progresses, but until the film’s climax John remains determined to drill into Homer’s head the realities of life as he has lived them:

Boy, you better take an interest in your own damn town. Instead of wastin’ your time worryin’ about Wernher von Braun and, uh, uh, Cape Canaveral.

John Hickam’s perception of reality is shared throughout Coalwood, with some exceptions. In the following exchange, Homer’s supportive teacher attempts to rebut Principal Turner’s attempts at setting her straight regarding career prospects in Coalwood:

Principal Turner: Miss Riley, our job is to give these kids an education.

Miss Riley: Mmm-hmm.

Principal Turner: Not false hopes.

Miss Riley: False hopes? Do you want me to sit quiet, let 'em breathe in coal dust the rest of their life?

Principal Turner: Miss Riley, once in a while . . . a lucky one . . . will get out on a football scholarship. The rest of 'em work in the mines.

Miss Riley: How 'bout I believe in the unlucky ones? Hmm? I have to, Mister Turner, I'd go out of my mind.

John Hickam’s whole life is the coal mine. As Homer’s sympathetic mother Elsie explains to Homer, the boy’s father loves the mine more, perhaps, than his own family. At one point, after Homer has begun to succumb to what appears an inevitable encounter with reality by working in the coal mine, John finally expresses his pride in his son. John is a heroic figure in October Sky, but one suffering from acute myopia regarding his family and their prospects for a better life. He expects Homer to work in the mine because that is the only reality he has ever known.

Why does Homer's father want him to work in the mine in "October Sky"?

As in so many impoverished areas, Homer's town has basically one place where a person can work and make a living.  For Homer, this place is the coal mine. In a sense, the men in the town feel a sense of destiny about going to the mines; there is no choice.  Homer's father feels this destiny, and thinks that what Homer does with rockets is simply a waste of time.  In a sense, it is not that he wants Homer to work in the mines as much as he feels that Homer should just accept the way things are rather than having empty dreams that will leave him greatly disappointed.  That is Homer's father believes in the old adage,

Blessed is he who expects nothing,/For he shall not be disappointed.

But, inspired by his teacher who gives him hope, Homer tries for the Science scholarship that will launch him into college and the future for which he hopes.

What did Homer's dad do during the last rocket launch in "October Sky", and why was it hard for him?

Homer's dad embracing his son and his dreams represented the final act that had proven Homer's validity.  Winning the Science Fair competition, meeting his inspiration, proving to everyone in the town that his dream had merit all seemed to be a distant second to his father attending the last rocket deployment, and detonating the rocket for flight.  As the rocket is in the air, Homer's father stands amazed at the sight and embraces his son in a manner that reflects a father's love and admiration for his son.  The reader gets the impression that for the first time, the father is able to feel and show appreciation for his son. This is something that had not been fully experienced until that moment.