Barring Champions, as they weren't around at the time Ultimate's roster was completed, Midna seemed to have the most going for her in terms of one-shots. Her character was popular enough to get an Amiibo atop Wolf Link, she came from (up until recently) Zelda's best selling game since OoT, she was one of the original playable characters in Hyrule Warriors, and she even got a spot on Super Mario Maker as a playable character. It seems pretty unfortunate that she got assist trophy status, but I'm glad she's at least represented.
I figure the next Zelda representative has a good chance of being indicated by a mix of good marks through polls, promotional/pop culture representation, amiibo representation, recurring appearances, playable representations, and ease-of-development.
Impa has excellent chances, being an original playable cast member of Hyrule Warriors, having a favored cult status going back at least as far as SSB4 (when the unfortunately fake leak about her came out), enjoyed considerable success in Reddit's SSB4 DLC polls, and has had 8 whopping canonical appearances throughout Zelda's history - which is, aside from one other character, probably more than anyone else yet to be represented in Smash. Unfortunately, she's yet to earn her own Amiibo. She's had 2 playable appearances so far, both from Hyrule Warriors. Can be a unique character, or an easy Sheik-echo.
Beast Ganon is a weird one. He's the series' original villain - probably the only character her with more seniority than Impa - although he's already somewhat represented by Ganondorf's Final Smash. Not quite the same, but similar enough to raise eyebrows. I'm unable to find much polling request for him, and he doesn't seem to be all too well represented on promotional websites like Play Nintendo or TV advertisements. Furthermore, he lacks amiibo representations, and as far as I can tell, any playable appearances. Would most likely need to be a unique fighter, as no Zelda representative appears to qualify as a good template for him. While everything seems to be going against him, he's arguably the most iconic Zelda character of anyone listed here, which is worth something.
Skull Kid is in a similarly weird place. He's had fame arguably at least as far back as Brawl, considering Sakurai clarified his non-playable status when his assist trophy was revealed, and the VGTribune victory. He has also consistently polled better than just about every other contending Zelda newcomer, and enjoys such coveted cult status as to be made the subject of various indie movies and licensed musical remixes. Unfortunately, he lacks Amiibo representation for now, and will likely have to be a unique fighter, considering he has no obvious Zelda counterpart to be made an echo of. Has had 2 playable appearances so far (both from Hyrule Warriors), and has appeared in 3 canonical games. Given his cult history with Smash, and the number of what are arguably teases from Sakurai about his inclusion, his chances seem strong.
Hilda's best chances are probably as an echo for Zelda or as an alternate costume - Koopa Kid or Alph style - all things considering. While a one-off, she enjoys considerable recognition right now, riding Link Between World's relatively recent fame. While she was unfortunately not playable in Hyrule Warriors (or anything else), she did gain the distinction of a unique alternate costume for Zelda. She's appeared in one game so far, and unfortunately never received an amiibo. I am unable to find mainstream pop references to her through online promotional material (apart from that advertising her game), or tv adverts, and I can find no substantial polling reference for her. Fortunately, she would be a very simple echo for Zelda, requiring minimal effort to complete. Who would have expected Dark Pit last game?
Tetra (or Toon Zelda) is a dark horse among the contenders. Sakurai initially tried to create Toon Zelda for Smash Bros. (and possibly Tetra), but was cut for unknown reasons. Since then, the two characters have enjoyed lasting recognition. Tetra and Toon Zelda both have consistently shown up in polls ever since Brawl days (Tetra moreso for America/EU, Toon Zelda moreso for Japan). Both of them have enjoyed wide representation outside of their own games, Toon Zelda appearing in Nintendo adverts in Japan, Tetra enjoying wide promotional popularity across media, even currently being featured on Play Nintendo's "Friends" section, which introduces the series to new gamers. Toon Zelda received an Amiibo, while Tetra has none yet. Both Tetra and Toon Zelda have had 3 playable appearances, each in both iterations of Hyrule Warriors, while TZ was playable in Spirit Tracks, and Tetra in Super Mario Maker. Tetra has had 3-4 canonical appearances (if you count Spirit Tracks), while Toon Zelda has had 5. Toon Zelda would make a relatively easy echo to Zelda, whereas Tetra could be a relatively easy Toon Link echo, or unique character. Given her mainstream appeal, her chances seem surprisingly good.
There are a few others, such as Twili Midna, and Tingle. Given the assist trophy thing with Midna, and the active role Tingle plays on Great Bay stage, I tend to doubt them, but there's always the possibility.
It's nice to see such a strong crop of contenders this year for Zelda representation! May the odds be ever in your favor, fanbases! Let us hope for many open slots!