Longer writeup by ReiToei has been removed for some reason, so mine might seem a little terse:
Superficially, Gay Byrne is a purveyor of light entertainment, with the bland persona of a million other chat show hosts, so his level of penetration into the Irish psyche and culture is perhaps difficult for the outside observer to appreciate. However, the following quote from U2's The Edge might help:
"The great thing about living in Ireland is that, no matter how famous you are, you can never be more famous than Gay Byrne".
Gay Byrne is not related to his namesake, the actor Gabriel Byrne.