The tinderbox is the closest thing to a lighter in the rules
If you are looking for explicit references to how to light a fire you are not going to find anything more than this.
This small container holds flint, fire steel, and tinder (usually dry cloth soaked in light oil) used to kindle a fire. Using it to light a torch - or anything else with abundant, exposed fuel - takes an action. Lighting any other fire takes 1 minute.
Making a lighter from the tinderbox
If you are interested in allowing a player to make a lighter in game and are looking for mechanics, I would recommend simply reskinning the tinderbox mechanics and call it a lighter.
Depending on how the player goes about making it (and how generous you want to be) you could even give it some advantages over the tinderbox. For example, you could make it waterproof or you could allow it to be used one-handed.
You could also model it after the Rock Gnome's tinker racial ability and just make it a mundane mechanical item:
Fire Starter. The device produces a miniature flame, which you can use to light a candle, torch, or campfire. Using the device requires your action.
Don't worry about fire-lighting mechanics unless it makes the story more interesting or fun!
Adventurers likely know many ways to start a fire: flint and steel, charred wool, etc. Lighting a fire is an essential adventuring activity and doing so would be second nature to most adventurers. For the most part, it is just assumed that adventurers are capable if not very good at lighting fires.
So, under normal circumstances, it doesn't seem super interesting to focus on and track the minutia involved in starting campfires and the like.
If you prefer to do this at your table, or if the characters are in a circumstance that does make it interesting, just use the tinderbox mechanics and you're done.
If the player is trying to use this to gain some sort of mechanical advantage (not just reflavoring a useful item) then you will have to judge to make sure it won't negatively affect your game. I don't see much potential for abuse however, especially if you require that lighting something on fire take an action at least (as per the tinderbox rules).