So, those of you who have read the novels by Rick Riordan are aware of the quality of them. It's a very interesting setting, one ripe for fan-fiction, at least on Webnovel. There is one that I enjoyed reading but the Author hasn't uploaded anything in a while. I figured if I want a novel written on this, the least I could do is offer an idea to get started with. I can never tell what works for people on this site, eventually one of these will be picked.
I was debating on who the MC's godly parent should be, but as evidenced by the title I decided on Zeus. I'll explain how later but let's talk about possible abilities first. I can't remember if Zeus's children had standardized abilities in the original story or if they wildly varied from person to person. My thoughts are that he should have both electrokinesis and aerokinesis alongside enhanced strength, durability, agility and reflexes, as well as decelerated aging after reaching maturity. Now, I'll go into specifically what is entailed by electro and aerokinesis. Before knowing of his heritage, he'll have no conscious control of these powers, they'll manifest as a higher resistance to shocks and protection from falls (that being said, it won't save him from falling from a building, it would more protect from everyday falls i.e. stairs and the like). After learning of his parentage, he'll connect the dots and begin to consciously control them, at this point he'll be limited to maybe small shocks and mustering up a light breeze. Once claimed these powers grow explosively, resulting in bolts of lightning and gliding through the air with ease. I'm just going with my gut for these powers, they seem like they make sense for a Sky God's child, there really isn't a precedence for this.
Since Percy Jackson feels calmer in water, I figured that the MC should have a similar trait, which would evolve into a hobby like Percy and swimming. The protagonist would feel at ease at higher elevations, which could lead to a few different hobbies like: Rock Climbing, Free-Running, Sky-diving, Hang-gliding, or just straight up flying a plane. Free-running seems to be the easiest to pick up with Rock-Climbing a close second.
Now let's talk about how Zeus manages to have a child outside of the Big Three's oath. The first possible answer I thought of was to make the MC Thalia's twin brother, and have him be sent to another family for care-taking, in order to prevent both from being found at the same time. This is an easy explanation as no new events really need to occur for this to happen, thus deviating only slightly from the original plot. Another possibility is to have Zeus go out and sire a second child in response to Percy's conception. The more believable choice is the twins, as it wouldn't take much to write that in.
If it wasn't obvious by now, I was aiming for an insert character, as reincarnation is something that should be reserved for a less serious story, mostly because it is usually accompanied by wishes. An MC in this setting wouldn't need to amass a large amount of resources to become strong, they mainly just have to have been born to a strong god or goddess, and then training will take care of the rest