Earlier editions of D&D had specific rules for poisons.
Any of these may inspire you for poisons in D&D 5e, but bear in mind that 5e's simpler ruleset is an intentional design decision to avoid the game being bogged down with unimportant detail.
In Original D&D, there were separate saving throws for different categories of special attack, one of which was "Death Ray or Poison". In D&D 5e terms, this essentially required that certain types of attacks required a more difficult saving throw than others. You could be poisoned by a creature attack, a poisoned trap, or unwittingly drinking a poison potion. A saving throw could allow you to take half damage. The rules are a little vague but no creature has a listed damage value for poison, and I get the impression that failing a save against poison meant instant death.
AD&D 1st edition had specific rules for assassins, who may use learn to craft and poison (DMG 20). A table has rules for different sorts of poison, divided by types, dealing damage or death on a failed save and half damage on a successful save, having specific price, onset time and method of application (e.g. ingested or otherwise).
D&D 3rd edition had particular rules where poisons typically deal ability score damage, sometimes permanently. Poisons take effect immediately with a certain effect. They have secondary effects which occur one minute after the initial effect, and separate saving throws are allowed against both the primary and secondary effect. You can poison yourself by accident when applying a dose of poison to a blade. Poison is too expensive to use cost-effectively.
D&D 5th edition abandons the idea of ability score damage and secondary effects with poisons. Creatures who use poison now have their own specific effects, as do poisons used. There is no longer an assassin class in the core rulebooks. There is a "poisoned" status effect but poisons also deal normal damage or have other effects.
In my estimation, it is simplest to use D&D 5e's poison rules, which appear on pages 257-258 of the Dungeon Master's Guide. The older editions' rules typically allow poison to bypass the ability of hit points to protect a character's life, which violates an implicit design principle of 5th edition.