This name is a synonym of Spergularia marina (L.) Besser.
The record derives from WCSP (in review)
(data supplied on 2012-03-23)
which reports it as a synonym with original publication details: Hannover. Mag. 18: 222 1780.
This name record (from WCSP (in review)) was not included in version 1 of The Plant List. It is possible the same name was included from a different data source, to check this you can search for Arenaria rubra in version 1 of The Plant List.