My son's name is Brian, and when we visited Ireland, his Irish relatives called him "Young Brian" in Gaelic, which was something like "Breenough". I would like to learn the correct spelling for both "Brian" and "Young Brian". Also, is there a Gaelic spelling for "Donna"?

Brian is still spelled Brian in Irish, but, as you've observed, is pronounced Breean.
Brian Óg is Young Brian - two different words.
Donna can't be spelled in Irish as it's an Italian name. As Hall is also not an Irish name (despite recently invented Gaelicisations like de Hail, de Hál, Mac Cathail srl.), I'm afraid your name in Irish would be... Donna Hall!