Getting into what could be expected from League of Legends itself, Jinx there is a heavy weapons artilarist sort of character that has multiple abilities that pin down opponents, and a passive that buffs her movement after getting a kill/assist, allowing her to chase opponents or run away after successfully taking someone down. She also is able to swap between a minigun and a rocket launcher for her main attack, so there's quite a bit of complexity here.
I think what I would do is make her down special Switcheroo!, which swaps between Pow-Pow (the minigun) and Fishbones (the rocket launcher). The weapons affect her normals: both have physical attacks that are kinda slow, but necessary when she has to cqc, and projectile attacks where she fires the weapon. Pow-Pow fires multiple projectiles, and has pretty low end lag in exchange for middling range for a projectile, while Fishbones fires a single exploding projectile with a lot of range and knockback, but is quite laggy, and can hurt Jinx if fired at point blank. The strategy here would be to use Pow-Pow to rack up damage in midrange, and Fishbones to secure K.O.s, or harrass opponents that hang out at long range. Jinx's smash attacks are laggy, but whenever she lands them, it procks Get Excited!, giving her a sizable boost to her movement speed for 10 seconds, or until she is grabbed. Get Excited! is also procked by K.O.ing opponents or Assist Trophies. If it procks while it's already active, it resets the timer.
Her neutral special would be Flame Chompers!, which acts like a mix between Snake's Grenade and C4. The Flame Chompers are grenades with metal teeth that explode a few seconds after they touch the ground. They're thrown in an arc that can be influenced by inputing a direction on the control stick, and they're items too, so Jinx can pick them up and throw them, which causes them to explode on impact. Like with Grenade, you can press the sheild button to drop one at your feet. While the Flame Chompers are on the ground, any character other than Jinx that steps on it get rooted in place, and then exploded, allowing some time for Jinx to position herself to follow up on the explosion. Opponents can also detonate the Flame Chompers early by hitting them, though they may the lack range to do so safely, and can pick them up, but doing so without triggering them is difficult. Jinx can only have 1 or 2 Flame Chompers out at a time depending on what's more fair.
Side special would be Zap!, a straight shooting electrical projectile that can be aimed up or down. It has a lot of startup lag, but on hit, it paralyzes opponents for an amount of time based on their %. Essentially, it's an aimable Paralyzer, but only the charged version.
And then her up special would be a rocket jump or something I dunno.
All in all she'd be a mid-long range zoner/trap hybrid character with bad cqc, but multiple tools to set up "gotcha" moments that allow you to deal big damage or set up K.O.s.