This is an interesting concept. I don't know the exact answer, but poked around and someone has written a book on the subject of Prominence. Based on the excerpts of this book by Adam Helman posted online, there seems to be several flavors of Prominence and a variety of criteria that can be chosen for each:
https://books.google.com/books?id=kr8AM-w8IFQC&pg=PT165&lpg=PT165&dq=prominence+island+parentage&sou...
Once a specific Prominence definition is chosen, it looks like it would be an interesting GIS challenge. For starters, you would need a Digital Elevation Model (DEM) of the complete extent of the project area, which would likely cover a vast area. For example, to derive the Prominence of Mount McKinley in Alaska, one typically would start with a DEM covering North, Central, and South America.
The general solution process appears to be identifying the highest point in your project area (Peak 1), then looking at the next highest (Peak 2) and finding the highest point of land (the col or saddle) that connects each. Then one checks the elevation distance between the col and Peak 2 to determine the Prominence of Peak 2, assuming the distance is more than your cutoff amount. For example, if you determine your cutoff will be "for a peak to be distinctive, it must be at least 500 feet higher than the surrounding terrain connecting other peaks", then the distance of less than the cutoff makes that peak invalid for consideration of having any Prominence. Then repeat a comparison of Peak 2 to Peak 3, then Peak 3 to 4, etc. Note that Peak 1 is a special situation - it does not normally get a Prominence, but if one is required it is typically its elevation compared to sea level.
An article in Wikipedia offers some hints at the analytical process:
Calculations and Mathematics:
When the key col for a peak is close to the peak itself, prominence is easily computed by hand using a topographic map. However, when the key col is far away, or when one wants to calculate the prominence of many peaks at once, a computer is quite useful. Edward Earl has written a program called WinProm which can be used to make such calculations, based on a Digital Elevation Model. The underlying mathematical theory is called "Surface Network Modeling," and is closely related to Morse Theory.
Topographic prominence - Wikipedia, the free encyclopedia
However, most of the links are dead.
Chris Donohue, GISP