My fiance is running a campaign and the players have the opportunity to obtain a wyrmling black dragon. Now black dragons are chaotic evil, but the monk of the group believes that it shouldn't be killed outright just due to the fact that its race tends to be evil. He wants to attempt to teach and raise it in goodly or rather, less evil ways, but the other players want a backup plan to change its alignment if worst comes to worst. The problem is, the DM doesn't see anything involving forced alignment changes. Is there a mechanical way to handle this or is it going to be DM decision to add something to the game that isn't currently there?
There are three ways to forcibly change something's alignment on the books. But none of them really work out well for you.
Forced attunement. Arguably the least dangerous option, if you can somehow force the dragon to attune a magical item you might change its alignment. Perhaps a Geas? If your DM sees that working, and you have any of The Book of Vile Darkness, The Eye of Vecna, or The Hand of Vecna on you, the dragon's alignment will change. To Neutral Evil.
To the Planes. Send your dragon on vacation to the Bytopia for 4 days ("Pervasive Goodwill" optional rule, DMB pp.59-60), The Abyss ("Vile Transformation" optional rule, DMG p.63) or the Nine Hells ("Pervasive Evil" optional rule, DMG p.64) and its alignment might change to Neutral Good, to Neutral Evil, or to Lawful Evil. (But dispel magic or remove curse will get rid of the Bytopian effect.)
Enlist the Slaadi. Unfortunately (?) this won't work on your dragon--only on humanoids. But in the interests of completeness, here's the third way to forcibly change something's alignment: true polymorph the dragon into a humanoid form, then get a Blue Slaad to bring your (humanoid) target to 0 HP. It'll automatically become Chaotic Neutral. Of course that's a consequence of it becoming a Red Slaad.
In all seriousness, parenting's a hard job.
Raising something to "be good" in a world where alignment's a thing written into a stat block is absolutely a project you and your DM are going to have to tackle together. There're no easy outs here. (But there are good story opportunities here!)
But it just got a little bit easier....
Now, a half-year late, I've got to challenge the frame of your question. You say "now black dragons are chaotic evil." And you probably got that from the stat block, or from a generalization you're making based on all the black dragons you've encountered. But
The alignment specified in a monster's stat block is the default. Feel free to depart from it and change a monster's alignment to suit the needs of your campaign. If you want a good-aligned green dragon... there's nothing stopping you. (MM. p.7, "Alignment)
We all know that rule zero empowers the GM to change anything, but this is one of those situations where a published book says "hey, GM: you might need to tweak this dial." I know this question was all about getting "plan B" in order, but I think "plan A" has a little more going for it than one might have assumed.
If you can trick or goad it to draw from a Deck of Many Things, it might draw the Balance card:
Balance: Your mind suffers a wrenching alteration, causing our alignment to change. Lawful becomes chaotic, good becomes evil and vice versa. If you are true neutral or unaligned, this card has no effect on you.
5E D&D is intended as a role playing game. You may be seeking RAW and 'Rules As Written' That said, if you want to play your game as RAF - Rules As Fun, you could make this into a brilliant role-play.
From experience: people gain many behaviours from biological circumstance. Be that from SAD (Seasonal Affective Disorder) that makes someone depressed, 'Roid Rage' when anabolic steroids cause people to act out in seeming rage or even any weird mood swings one can gain from any kind of malnutrition: the body has a strong input on feelings, thoughts and behaviour.
I recommend you role-play your Black Dragon as a creature with these odd 'chaotic and short sighted' behaviours. If one suggests a 'good' person is compliant with group needs, this dragon would be of a body that would not allow it to make full use of its extremely high wisdom and intelligence scores.
If you want a simple model for how this creature might behave: imagine an abusive relationship. This Black Dragon may genuinely love the parents! This would not stop them from being possessive, terse, jealous, greedy and cantankerous. This can make for some fantastic situations in-game. You can imagine that after years of terrible dragon-child rearing experiences the 'parents' eventually give up and leave the beast be. Perhaps the dragon suffers from some version of PTSD or even RAD ('Radical Attachment Disorder') - but the DM would still get to choose how this dragon would react.
A clever DM can make a loyal and loving motivation set without losing the chaotic and evil part. The best example may be Christine, the 1958 Plymouth Fury from a Stephen King book (by the name name). She was a terrible communicator and tended to slaughter anyone and everyone - but did her best to 'protect' her owner (though she failed).
No, you really can't change essential nature. Alignment indicates tendencies, not absolutes. You can train the wyvern to do good, but it's nature will always steer it to thinking selfishly for itself. Any training and listening will not be loyalty based, but based on fear of consequences and tangible rewards for loyalty.