This post was written January 17, but it won’t be posted until the 29th. I’ve only made small edits since its initial draft.
Leaf recently wrote a post outlining what he thinks are the “sleeper” amiibo that will make an impact as the meta continues. I read through it and had some additions and reinterpretations to include.
I think these amiibo are the most likely to become an influencing force in the metagame. They won’t necessarily become top contenders, but they will influence the lineup of top contenders in some way. They could become a hard counter to an amiibo that would otherwise be a tournament winner, or they might sweep out the lower tiers more consistently than a top tier amiibo would.
Jigglypuff

I agreed with Leaf’s interpretation that Jiggs’ weakness was her complete lack of mass. I think this fact will hinder her from ever being an S tier comptetitor, Jigglypuff’s new AI will allow her to put up quite a fight against large, hard-hitting fighters like Bowser and Link. The fact that she can now use aerials competently means she may be able to learn how to chain aerials to force an opponent offstage, and the heavyweights have an even harder time recovering thanks to Ultimate’s new ledges.
Her Rest is now usable, which in itself means she has much more potential. Rest being in play gives Jigglypuff a raw KO move that she didn’t have before, and raw KO potential is what leads the amiibo scene. As training methods develop and become more effective at getting the desired moves, JIgglypuff can only become a better option.

Mii Gunner
You’ll see what I mean once you see ASMR #2’s videos, but Vintro’s Mii Gunner presents the possibility that projectile spam could once again return the meta to a ground-based game. In the ASMR matches that his Gunner played, he basically stayed on the ground without jumping at all, and spammed the Mii Gunner version of Samus’ missiles to carry his opponent offstage. This worked incredibly well until his opponent got in close… at which he used a tilt or smash attack to get them away again. Whatever training this Gunner went through, if it works for another amiibo then it could change quite a bit.

Young Link
Most Young Link trainers have an issue with their Young Link suiciding occasionally. That’s a pretty big problem, considering the rarity of stocks in most rulesets. In ASMR #1, a Young Link was submitted that didn’t have this problem: in fact, he was so competent he nearly took a set off the Link that won the tournament. They’re similar characters, which works in Young Link’s favor, but Young Link hasn’t seen much tournament representation.
Whatever this character has to offer hasn’t been seen yet, and it definitely hasn’t been factored into the series of choices surrounding the metagame. It’s possible Young Link may emerge as a Link counter at some point in the future, and my early research suggests this as well.

Doctor Mario
In Smash 4, Mario was one of the best amiibo in the vanilla metagame. His AI was flawed, but his character was so good there was just no getting around him. He was the only character besides Bowser to put up a solid fight against Ness, and he also countered Bowser. Mario was a great choice for any amiibo trainer.
Then Mario sunk into alcoholism and really let himself go, and it shows. He’s been heavily nerfed since Smash 4, and his AI seems like an actual drunk is controlling him. Mario’s not good anymore. His big brother, Dr. Mario, has chosen a separate path.
Doc has decided to fix himself up and hit the books, and it shows. He’s got most of the tools that Mario has, but he’s smarter and more precise. His down tilt is sometimes (seemingly accidentally) used as a combo starter into his down air or forward air. Just the fact that he has that option means he’s got a little bit of leverage against opponents on the ledge.
Offstage, Doc has his down air for a meteor smash, and his forward air for a blastzone KO. His recovery is somewhat lacking, but his Tornado can be used to interrupt an opponent’s recovery while bringing him closer to the stage.
His onstage play is somewhat similar, but with the addition of his Cape he’s now immune to projectiles as well. This spells doom for characters that are projectile-reliant like Samus, Mii Gunner and to a lesser extent Link. His smashes also hit hard, despite their very limited range, which places Doc in the unique position of requiring up-close combat for both him and his opponent.
As far as I can tell, his one weakness will be melee characters with long ranges like Lucina or Bowser, which will keep Doc from rising to the S tier rankings.
In conclusion…
At present standing these amiibo aren’t very useful or competitive, but I believe that they have the potential to become a meta-changing fighter. When our training methods improve and we get better behaviors out of our amiibo, it seems to me that these characters are the most likely to make a difference.