You can find "Reier"s and "Brynildsen"s in Rogaland -- but actually I don't see any proof that their father had to be from Rogaland, consider that too.
Remember to allow for the spelling variations -- Reier, Reiar, Reyer, etc.... but you would probably expect the name to START WITH Re... at a minimum.
There are also variations on "Brynildsen"s but you would expect them to at least START WITH Br... Luckily there aren't a large number of male Norwegian names that start with Br...
That original Hetland parish marriage record for Malene appeared to list the Rogaland area of Høyland/Høiland? and the 1865 census appears to list a farm then by the name of Gjelle or very similar. Might be worth investigation. That record also gives some verification for your mother's suggestion of a Reier Brynildsen.
Almost all original records of emigration departures from the port of Stavanger were lost a in fire years ago. Those available in databases online at this time are merely being rebuilt from other sources as available -- many emigrations no longer documented from that port.