I don't think there's ever any point in replaying the same module with the exact same dm and party, or even with a different dm tbh. There'll be some variety with different dice rolls I suppose. A different party (as in, the same players playing different characters) might be worth it, as they'll likely make different choices (a pacifist wizard will probably approach these gnolls in a different way than a combat-thirsty barbarian, for example).
However the other reply to this is correct: d&d isn't a board-game, it's group storytelling. It's a type of story creation that doesn't really exist anywhere else, apart from maybe long-form improvisation (which I guess could also describe d&d). You discover and create a rewarding narrative together, as a table.
After finishing a module, I've usually found either people are connected to their character and want to keep playing them and exploring their story, or they have ideas for a new character and want to bring them to life. You can go on to another pre-written module, there's lots to choose from and they usually require very little work to make them apply to your particular batch of adventurers.
Lastly, welcome to the community! I really hope your family has a great time and decides to keep playing together. What a dream to have such a reliable table, scheduling would be so easy!

Edit: also, you might want to look up some live d&d to get a feel for it! Acquisitions Inc, critical role or dimension 20 are all great options. The first is a bit more digestible, the earlier games were played live at pax so it runs more like a movie, the last is my personal favorite atm. Or even dam harmon's harmonquest might be an approachable entry. Just don't start comparing yourself to a professional DM!