You don't need homebrew for this, because 5e already accounts for using ability scores for things that aren't explicitly covered in the non-exhaustive list of skills (appropriately enough, this is first touched on in the "Using Ability Scores" section of the Player's Basic Rules, where each ability non-exhaustively lists examples of Other X Checks that the DM could call for).
You make a ruling instead, and move on with the game.
For those not familiar with such improvisational freedom in the DM's role, here's a basic breakdown of how to make a ruling for this kind of action:
- Pick an stat that is relevant to what is being attempted and how it is being done.
- Set a DC for the attempt, based on how hard it would be for an average untrained person to accomplish given the current circumstances, materials, rush the PC is in, etc. (You can refer to PBR page 58 for a list of DCs by subjective difficulty.)
Yes or No: does the PC have a relevant skill?
- If no, skills are not relevant. Have them roll their ability against the DC.
- If yes, they get their proficiency bonus as normal.
Have them roll against the DC, adding whatever relevant modifiers (e.g., proficiency) to the roll.
DMs used to making rulings like this (read as: "using the existing rules to handle an unexpected action") don't consciously go through this whole process, but it's an accurate description of how it effectively works. In practice, and with practice, this becomes second nature and doesn't have to be explained and laid out like this.
For example, I would have the character make an Int check, since jerry-rigging an explosive arrow relies mostly on coming up with an idea that works and being clever enough to figure out how to execute on it properly.* A relevant skill† might be Improvised Explosives, Knowledge: Explosives, or even I Have A Cunning Plan (though that perhaps would cause them Disadvantage...). I'd pick a DC of, oh... let's say "Hard" (DC 20), and inform the player of such. Then I'd ask them if they'd like to proceed, and have them roll if so.
But that's just how I would make a hypothetical ruling. I might make it differently, based on the exact circumstances and methods involved at the moment. The point is, you can't really do it wrong, except by letting it bog the game down. Make a ruling that's good enough, and move forward to get to the fun stuff. Consistency evolves naturally as you establish a history of rulings and they guide your feel for how to make new rulings.
* If they were also trying to do it in a rush or otherwise being reckless, I might have them make a Dex-based saving throw (in addition!) to not blow something up, but skip that if they were taking time to be careful.
† Skills are usually listed with their associated stat, but they can be used with other stats if the circumstances make sense (e.g., Sleight of Hand is normally (Dex), but could be used to judge a friend's outfit for vulnerability to pick-pocketing using Int). As such, I haven't bothered to list a stat with these made-up skills, because they fit the circumstances just by the nature of the way this example is constructed.