I am the DM in a fairly new d&d 5e campaign. In it is a kobold barbarian who was once a dragon-shield in a now destroyed tribe. The black dragon wyrmling the tribe worshiped/protected is presumed dead by him. Another character is a dragonborn paladin of Bahamut who is taking the oath of redemption for a violent nature he struggles to control. Both characters have been bonding a lot over their past failures and desire to be better.

I had an idea that the party could find the wyrmling at some later point, having been imprisoned by someone connected to the campaign's BBEG. They would then have the opportunity to protect the dragon from the BBEG and share their conviction to be "good" with the dragon who would have had an evil tendency otherwise. As a note I tend to treat alignments in sentient mortal creatures as changeable to varying degrees.

Is this a bad idea from a storytelling perspective to give this opportunity?

It is very good storytelling.

One of my top tips for DMs is, if you can shove player's past into the mix and make it look flawlessly pre-meditated, the players will adore you. It is always good to have the player's backstories come haunt them enough to push them into action but little enough as to not make an entire campaign centered around two of these characters.

But that doesn't mean that the spotlight can't shine over one or two players every now and then. That and, hey! It lets them also integrate with each other, and a party that's integrated is a party that will participate with the story much better and runs less risk of splitting up.

A general rule of storytelling is that you can go wild with unlikely random coincidences to set up the story (first act), but while the story is progressing (second act) they should be used more sparingly. The finale of the story (third act) should be the logical and deterministic conclusion of the events which happened before.

Unlikely circumstances are found at the start of many great stories. For example:

1. The Lord of the Rings starts with a simple peasant gaining
   possession of the most powerful magic artifact in the world.


2. 
   

   The Chronicles of Narnia start out with a group of war refugee
   children finding an interdimensional portal in a simple wardrobe.

   
   
   

   Likely? No. Interesting? Definitely.

   
   

The reason why this works so well is that unlikely coincidences allow you to create situations which are interesting because they are unusual. The audience (or in your case the player group) gets excited to know how this unusual scenario would play out.

An unlikely coincidence which changes the story in the second act often leads to a "plot twist". This is a double-edged sword. A good plot twist can be interesting, but audiences might also describe them as "contrived" if they feel that it breaks with the premise of the story they were excited to experience.

An unlikely coincidence which resolves the story is often referred to as a deus ex machina (if it leads to a happy end) or diabolo ex machina (if it leads to a bad end). It is often considered lazy writing, because it deprives the audience from finding out how the story would actually have ended if the author would have allowed for it to end naturally.

Now let's apply this to the situation from the question.

Stumbling upon the wyrmling which is connected to the character's past, and presumed dead, in a completely random and unexpected situation would be an acceptable coincidence if it happened right in the beginning of the campaign.

If you do it in the middle of the campaign, it might seem contrived if it happened in a very unlikely situation. You might want to foreshadow the encounter. Drop some hints that the wyrmling might still be alive, that the BBEG is somehow connected to it, and that it might have some connection to the place where it is eventually found.

If you make the wyrmling appear our of nowhere in the final battle and help defeat the BBEG without any setup whatsoever, then that would be lazy writing and the players will likely feel disappointed.

It's great. Put it in.

Stories, ranging from novels to comics to movies and tv shows are full of necessary "coincidence." And so is life. As a GM surprise connections are important to me and my players. The question is not so much should you include the coincidence, but how do you build up to it and make it believable?
Several techniques you can use to strengthen a narrative coincidence, and gaming narrative in general, are:

• 
  Rich world building: Develop a world that feels alive. Create history, characters, societies and cultures that surround the players and support the story you are teling. You will probably create quite a bit that the players never hear about, but it will inform your npc's behavior and the details of city walls, tapestries, scrolls and other minutia that the pc's come across. I find world building to be fun and essential to my campaigns. If I know the history and culture of an NPC's city they become a real person and are very easy to improvise.


• 
  Hints (foreshadowing): Create clues that indicate there may be an unexpected coincidence or surprise, clues should be subtle, not obvious indicators. Clues can be events or situations they will look back on and think, "yes this all makes sense now!" Clues need to be thought out and planned.


• 
  Sense of free will: If I try to control the narrative too much my players will pick up on that and feel like they are pawns in a game, rather than living breathing characters with free will. So whatever happens, the players should feel like they reached that goal / attained that knowledge through their own actions, not because it was something you had preordained. This comes from regularly presenting to your PC's a variety of choices that will have very different outcomes, and also letting your PC's think creatively and make decisions that you may not have anticipated. Some of the best moments in my campaigns happen when the players take a wild or creative action and I have to come up with some good story telling on the fly.


• 
  Believable Dialogue: Dialogue can go in a lot of directions, it can be sinister, silly, comical or serious. Whatever the tone is, the players have to believe what the character is saying and it must be grounded in the reality of the game. If they meet the BBEG at the end and he's all like, "Ha! I've had you're little wyrmling all along! I've led you here to taunt you!" That is terribly obvious and dull. But if instead the BBEG's henchman looks at the barbarian and says, "You. You bear the mark of the gladstone tribe. (evil stare) Was not that tribe vanquished years ago upon the plains of Ardour?" and a conversation ensues and secrets are revealed, etc. The BBEG is as surprised as the PC to discover he is from the black wyrmling's tribe, and this sets up the showdown. It also makes the coincidence more real and it doesn't feel preordained or forced.

All of these should somehow connect the wyrmling, the barbarian and the BBEG (or thug thereof) with a narrative thread that is woven throughout your campaign. Build up to it, lay hints, create a richly textured world, give your PC's free will and make the reveal natural. The best way to learn is to do the best you can and then get feedback from the players. You totally got this.

PHB 312 and DMG 316 have long lists of books that can help with narrative and world building.

No Easy Answers

Even restricting to science fiction and fantasy genres, you could write volumes on the role of coincidence in good or bad narratives. And then you'd have to disregard a lot of that work because RPGs are not narratives in the same sense as novels or movies.

Not to mention, there is a fundamental matter of taste and style from player to player.

Therefore, I tend to treat coincidence as a scarce resource, which renews itself slowly if at all. And here's the kicker: I, as a GM, never know how close to the edge I'm dancing, because the ultimate authority is the audience, your players.

What Is The Purpose Of The Coincidence?

That said, I try to answer that question-- what is the purpose?-- to try to evaluate whether the coincidence is worth spending, so to speak. Player/character agency is central to modern gaming, so that is the lens I try to use.

Is this a coincidence that will set one or more players up for an interesting choice? A revealing choice? A choice that will define their characters or have manifest impact on the game as it goes forward? If yes, then it's probably a good coincidence, a coincidence that leads to agency.

Is this a coincidence that just gets the characters to do what I want them to do, or be where I want them to be? If yes, then it's probably a bad coincidence, because it is restricting their agency.

These are not absolute, of course: Sometimes, players use their agency to do something really catastrophically dumb and a quick coincidence saves the game. (After all, it's hard for anyone to have agency if Sauron wins.)

And even for those good coincidences, there is a built in tension: If you are throwing so many coincidences at the players that they see one every other session, that starts to feel like a systemic reduction in agency because they're probably not reaping the full rewards (narratively, anyway-- sometimes a narrative reward means hard luck for the character) of the choices they made a session or two ago.

So Ask Yourself This:

What is the point of having them run into this wyrmling again? (And indeed, a more philosophical question is: What is the point of even having character backgrounds?)

Is it to illuminate something about the characters? To cause them to reveal something about themselves? To force a difficult choice? If it isn't... can you tweak the idea until it is?

I tend to believe in two coincidences/random picks being good storytelling, any more and it starts getting contrived. In your example, the Kobold's tribe worshipped a wyrmling (coincidence A that it's wyrmling X) and the BBEG has a wyrmling (coincidence B that it's wyrmling X). If the dragonborn also had some relation to the same wyrmling, that'd be a third coincidence, and be pushing it.