To answer your original question... I do not know of any specific mechanics, feats or whatnot in 5E or any previous edition that deal with the creation of slimes. The most likely place to find something of this sort would be an old edition of Dragon Magazine, as they typically have all the weird rules that some random writer invents for an article... but I couldn't name a specific article or issue that deals with it. Sorry.
That being said... as far as ideas for established slime-related abilities that you could incorporate into your game, I'd say go looking for any bits and bobs related to the demon lord Juiblex, the Faceless One. It is the master of all slime-related stuff, so you could probably find cultists that have slime-related features to steal, rituals and the like connected to the demon lord etc. For 5E you can look at Out of the Abyss to see if it has Juiblex-related material... and from 3E I know the Book of Vile Darkness also has Juiblex/demon info in it.
That being said... as far as ideas for established slime-related abilities that you could incorporate into your game, I'd say go looking for any bits and bobs related to the demon lord Juiblex, the Faceless One. It is the master of all slime-related stuff, so you could probably find cultists that have slime-related features to steal, rituals and the like connected to the demon lord etc. For 5E you can look at Out of the Abyss to see if it has Juiblex-related material... and from 3E I know the Book of Vile Darkness also has Juiblex/demon info in it.