My favorite Disney film as a kid was Aladdin. I'd actually been familiar with the story from 1,001 Arabian Nights prior to its release in 1992, when I was six years old, but as is often the case with Disney adaptations of classic fairy tales and similar legends, the movie became my main touchstone with the story.
Naturally, the biggest draw to me was Robin Williams' Genie, whose fast-talking, well, Robin Williams-ness, was exactly the sort of manic energy that would send deep roots into a young kid's mind. The Genie is profoundly likable, and there's also something deeply sad under it all where his over-the-top performative behavior is all part of a desperate attempt to get Aladdin to like him enough to grant him freedom.
Most fantasy in Western culture is, unsurprisingly, based on folklore that comes from Europe, but the djinn have a long and storied role in Middle Eastern folklore, and like most folklore creatures, they take on different roles depending on the writer. In some cases, the djinn were wicked spirits, but the stories I always found more interesting is that they were simply another humanoid people. In some stories, they are made of smokeless fire, and I believe some interpretations have them being made of fire in the same way that humanity was, in the Genesis story, created out of earth.
In D&D lore, they take this idea of Genies being tied to elements and run with it, making them elemental denizens of the elemental planes.
That already makes them fairly unique. Most elementals in D&D (and fantasy in general) are barely sapient, mostly just living embodiment of aspects of nature, most classically divided into the four classical elements of Western alchemy.
The Genies of D&D are divided similarly, but they are far more human-like than other elementals we encounter (with a few exceptions like Azer.) Evidently, genies in D&D are human-like enough that they can have children with humanoids, which is how we get the Genasi playable race.
The lore in the 5th Edition Monster Manual (I'm just going to set aside the thing about them all having slaves - while I get that this is in part to show how different their worldview is from the mortal races, it carries some unfortunate cultural biases and stereotypes that I'd suggest downplaying - basically, if you want to get into slavery as a theme in a campaign, you can, but do so thoughtfully and bearing in mind the implications of what you put out there) gives the following origin for genies: when the soul of a sentient creature melds with the primordial matter of an elemental plane.
The soul is altered enough that the genie doesn't remember any previous life. That being said, the soul will determine aspects of the genie's personality and to an extent their appearance. Genies are only formed this way, and don't have their own genie parents, but this connection to life is what allows them to sire genasi offspring with mortals - though that's said to be quite rare.
Opulence is a big deal for genies, and seems to be a major way that they establish their power within genie society. Having the most ornate palace with the most servants (voluntary or otherwise, I guess) grants them status.
Tasha's Cauldron of Everything introduced the Genie patron for Warlocks. I actually love this as an option (damn it, there are like four Warlock characters I want to play... actually yes, that's literally true) given that Genies run the gamut of alignments. While the Monster Manual gives specific alignments for its four genie types, these are not beings of the Outer Planes, and so I'd say that you could probably find a broad variety of alignments amongst them. I think you could say that they might have personality traits that are thematic with their element - the air-based djinni (which is a little confusing given that "genie" is just an anglicized spelling of djinni, but in D&D every djinni is a genie but not all genies are djinn) might be, well, flighty and shifty, or the fire-based efreet might be passionate and determined.
Anyway, as a patron, a genie might have complex plans, or they might be simply interested in obtaining greater wealth. Frankly, I could imagine that a genie might form a pact with a warlock purely in the interest of having a powerful adventurer that they can claim as their servant.
One of the most potent reasons for bringing a Genie into a campaign is that they can grant wishes to other creatures - providing access to the Wish spell. Wish is explicitly the most powerful spell in the game, but players will only ever see it at level 17 and up (and then only among a few classes).
While the practical adventurer's use of Wish is typically just a free "I can cast any 8th or lower level spell with a single action and no expensive components required," Wish can often be crucial to rescue characters who are beyond the limits of typical resurrection or restoration magic. If a player character is turned by a vampire, for example, you need a Wish spell to turn them back without killing them and then reviving them using either True Resurrection (which is itself a 9th level spell) or Reincarnate (which you could argue doesn't really work.)
A genie could help out of benevolent intent or in exchange for something, but you could make an adventure of finding one that is powerful enough to grant a wish. Also, the genie is capable of interpreting the wish's wording, which could lead to unintended consquences.
This differs from a deal with a devil, though - a devil is fundamentally evil. Genies aren't inherently evil. So, while a devil is never going to agree to a deal that doesn't allow them to do some greater evil (even if you word it carefully enough that you aren't the recipient of that evil,) the party won't necessarily know if the genie they're dealing with will give a generous and benevolent interpretation or a cruel and/or self-serving one.
While genies are mostly known for granting wishes to mortals, I don't think there's anything that explicitly says they can't use them for themselves. Clearly, there's some limit, either on power or imagination, that prevents ambitious genies from just taking over the universe with these spells, but genies could easily serve as the "man behind the man" kind of villain. Maybe there's some warlord who suddenly takes over a kingdom and becomes insanely powerful out of the blue. The party could seek to investigate and discover that they've been able to do so with the help of a genie.
A genie can also serve as the type of villain who's less cruel and more callous - simply seeking to steal and loot in order to add to their riches. You might thus have an antagonist who's not so much an embodiment of pure evil, but simply a criminal leaving chaos in their wake as they rob the material plane of riches.
Classically, genies in folklore often offer to grant wishes in exchange for being released from a vessel that holds them. One could imagine that a genie that spent ages trapped in some tiny vessel might seek to bring ruin upon the mortal world, or maybe more specifically a part of the world where they were trapped. In some Islamic traditions, King Solomon was able to control djinn, and I could imagine taking that as inspiration to imagine some legendary wise ruler from a past era who captured a powerful genie (or many) and trapped them in jars or lamps or what-have-you. Maybe that leader is long-dead, but the genies that they captured would love to take vengeance on that leader's kingdom.
Now, let's talk parents.
Genasi are the offspring of mortals (typically humans) and a genie. While D&D has plenty of hybrid races, like half elves and half orcs, along with many humanoid races that are distantly related to other magical beings, like Dragonborn or Tieflings, I think Genasi are the only ones who are most typically the offspring of a humanoid and an extraplanar magical being. (Granted, you could easily flavor a tiefling as being the literal child of a fiend and a mortal or an aasimar as the child of a celestial and a mortal, but that's not generally how the lore is written.)
Having a genie parent must be very, very weird. Genies, as we saw earlier, don't have parents themselves. And because of that, they don't have siblings either. That means that genies typically don't have any families (at least not by blood.) They're also ageless. A genasi character might have siblings, but they have at least one parent for whom this whole "family" thing is a totally new concept.
I imagine most genasi don't know their genie parent, and might never have even visited the elemental planes. Thus, a character's quest could be to track down their absent parent (the air genasi druid in my first campaign was left in a basket on her human father's farm, so they don't all have to be absentee dads,) which would require becoming a big enough adventurer to travel across planes.
That being said, I think that having dead or missing parents is such a common trope for RPG characters that it's fun to subvert it. Your genie parent might be super-enthusiastic about being a parent (in part because it makes them unusual among their kind,) and if they're a kind and benevolent genie, they might even have an advantage in not carrying their own family baggage - able to focus their affections and attentions on their child.
In fact, my concept for a Genie warlock is actually to play a Genasi character whose patron is his dad (I generally like the aesthetic of the air genasi, but it's hard to turn down a fire genasi's darkvision and an efreeti warlock's access to fireball,) and in contrast with many warlock patrons, the two would have both a close and communicative relationship and also a really affectionate one.
There are only four stat blocks for genies in 5E, all of whom are Large, CR 11 Elementals. But I think that they're brimming with enough really interesting personality that you can do a lot with them.