The plaques with names on Ellis Island are there because someone made a donation towards the restoration. That adds another possibility for why one family member may appear and another not be shown at all or possible be separated from the first.
To search for your ancestor on Ellis Island's records you can start with their website (link below). I do not know if you can search their records for free but it is certainly worth a try.
https://www.libertyellisfoundation.org/
Not all the records have been transcribed so you may not find records of your ancestors' immigration online. Also keep in mind that not everyone came through Ellis Island when they entered the U.S. A few other large points of entry included Baltimore, Philadelphia, Boston, and Castle Garden (NY).
My wife is an amateur genealogist. David, if you get stuck, send me a PM and I'll forward what information you have to my wife.