Beast empathy. You can't actually communicate with animals the way a Parseltongue can, but you can more easily understand the intentions and habits of animals and magical beasts. Basically the kind of thing you'd imagine Newt Scamander had to keep the peace between all the creatures within his suitcase, as well as how he was able to learn so much about so many creatures within his lifetime.

 

Sex and tantric magic abilities ofc. The world demands a reason for the Weasley's heinous levels of fertility.

 

The Ability to physically turn into an Armchair and keep your Human Mind. Like the Animagus transformation but specifically to turn into an Armchair, though with training you may decide if you dont want your clothes to transform with you.

This is based on Harry and Dumbledore finding an Armchair which turns out to be Slughorn, my personal interpretation is that Slughorn just used magic to alter the way people perceive him(like Hogwarts makes itself look like a ruin to muggles), but most people seem to have interpreted it as him transfiguring himself, despite Slughorn being capable of thinking, feeling pain, talking and being able to become human again despite his supposed lack of a brain. If it was transfiguration the only explanation is that Slughorn is secretly an Archmchairmagus.

 

Canon also has Fleur's Veela allure ability, whatever you want to call that.

 

I thought Fleur was just super hot, it was only the veela at the World Cup and even then it was only when they danced.

 

She still got some magic aiding her, like she literally emits a silver glow(sometimes described as faint, other times as strong), while full Veela are mentioned to to have skin that shines brightly. It't mentioned that it usally makes others appear less attractive, but during her wedding it made everyone it hit look more beatuiful to the point that you didn't even notice Bill's scars. It's also implied that she accidentally affected Ron with her magic when she was trying to charm Cedric.

this is the description of the glow of Veela

and this is how Fleur is often described

and this Fleur durring her Wedding.

Her apparently affecting Ron

Also the Veela didn't just affect people when Dancing

 

In bad Naruto fanfiction, there's a pattern where everything is a bloodline. Characters will 'discover' new bloodline when they need a new ability and normal jutsu are just ignored. I remember one author note that said something like, "I may decide to give Sakura a second bloodline. Haven't decided yet'. It just struck me as incredibly wrong.

I think this suggestion (along with some of the others in this thread) are verging into this territory. Not everything needs to be a initiate ability. Tantric magic can just be something anyone can learn. Harry Potter is a story of a magic school - power through learning. Gating off something as wide as 'tantric magic' behind an innate ability requirment feels wrong.

 

I'll go with a rather simple one, a natural born Occlumens.

 

An affinity for the more physical bits of magic, potions, wands, etc. I'm not sure how it'd manifest - most likely just as a very basic form of intuition for whatever you're doing. Not a step by step guide, just "Hey, maybe I shouldn't drink that or add this to that". Might be common sense, might be a magical sixth sense. Though if it was extensive enough to apply to something like Wandlore, I suppose you'd never realise you had it. Your sense for what was 'good' wood would just be something you assumed you had developed over time as your training in the craft progressed.

 

The Ability to speak with magical beings in their own language. Basically parseltongue, but for creatures like Giants and Goblins.

I think this is even implied in canon, as the wizarding world article on wand woods mentions that Applewood users often have the rare ability to do so.

https://www.wizardingworld.com/writing-by-jk-rowling/wand-woods
Also looking at it, I think both Dumbledore and Barty Crouch Sr would be reasonable to assume have had both the ability and originally wielded Applewood wands.

 

One thing you could do is have lots and lots of very minor innate abilities as the result of charms or curses still kicking around from the distant past. Werewolf descendants who can't sleep at full moon have some precedent in canon, but a descendant of Ariadne who can always point to the exit seems like a plausible addition. Or you could add a pureblood wizard cursed to marry a squib by a rival, whose great-grandchildren are still cursed to stumble over their words on dates with witches/wizards (or at least, believe they are with much the same effect).

This gives a good reason to have any character afflicted/benefitting from an array of minor magics and ups the "background magical feel" a bit. Gives some justification for snobbishness about heritage/marriage if you want to go that way - a sort of "never know what they're carrying" view on non-established lineages.