Hello Joan,
Generally, if a coordinate reference system (CRS) is added to the EPSG Geodetic Parameter Registry, Esri will add it to the software. I'm on the subcommittee that maintains the EPSG registry so I will pass on information from distributors and customers upon request.
Getting the information into the EPSG registry is useful because it's very public and available for other software besides Esri software.
Here's some of the information that EPSG will request beyond what you might expect:
shapefile or coordinates outlining the area of usage. This is particularly important if the new CRS is usable both onshore and offshore.
Scope: ex. cadastral, engineering, surveying, etc.
Anchor realization (fundamental point or ITRFxx)
Datum epoch date
I have an Excel spreadsheet that has sample data submission forms for the various types of objects that EPSG supports. You'll likely get one from the Chair if you submit a change request. If you'd like to see the form sooner, please send me an email, mkennedy at esri dot com.
EDIT: The Change Request action on the EPSG website requires that you create a login. The login is not used beyond sending you notifications of updates to the registry if you opt-in. If you don't want to do that, contact me or technical support / international distributor and requests for new datums, coordinate reference systems, transformations, etc. will get to me and I can liaise with EPSG.
Although the EPSG name has been kept, the registry is now maintained by the Geodesy subcommittee of the Geomatics committee of the IOGP (International Association of Oil and Gas Producers).
Melita