If I understand you right, you want to the names of songs people are likely to know, so you are not conducting a silent singalong and feeling like an idiot.

Of course, much depends on the age of the group. People age 70 are going to know a lot of Irish songs. People age 10, who have spent much of their conscious lives playing computer games, are not.

Anyway, how about:

Danny Boy
Molly Malone (cockles and mussels)?
the band played on
My wild irish rose
sweet rosie o'grady
who threw the overalls in mistress murphy's chowder
morning is broken

Another source is church music

Be Thou My Vision,aka Lord of All Hopefulness
Anything using "The Star of the County Down" - yes, it's not really Irish, but so what
anything using the tune 'Nettleton'
Gather Us In - sounds Irish enough