There is a wiki page on the stan-dev/stan repository titled Contributing New Functions to Stan. It is out of date and not in a relevant location any more, so I think it needs to be dramatically rewritten or deleted.
The Stan Math documentation features a getting started guide with an example function: Stan Math Library: Getting Started Guide
We recently added a new page to the stanc3 documentation on exposing a new function: exposing_new_functions (stanc.exposing_new_functions)
I think whatever information we wish to say about adding new functions lives in one of those two places, not the stan-dev/stan wiki.