It is a made up American name and has nothing to do with Welsh. It started in America when Irish immigrants began to cherish and romanticise their heritage (when they were safely out of the manual-labour-straight-off-the-boat category of immigrants and nicely ensconced in the middle class). Other names in the same category are Tara and Shannon, and Colleen (which arose from a half remembered Irish word cailin, meaning girl). Alannah is another one (from the endearment (vocative case) 'A leanbh' - 'dear child'/'sweetiepie').
It's an anglicisation of the word Eireann, which is the genitive case of the word Eire.
You can find it in the hymn 'Hail Glorious Saint Patrick' --
'Thy people, now exiles on many a shore,
Shall love and revere thee till time be no more,
And the fire thou hast kindled shall ever burn bright,
Its warmth undiminished, undying its light.
On Erin's green valleys, on Erin's green valleys,
On Erin's green valleys look down in thy love.
Ever bless and defend the sweet land of our birth,
Where the shamrock still blooms as when thou wert on earth,
Our hearts shall still burn, wheresover we roam,
For God and Saint Patrick, and our native home.' (iirc)
Irish people tend not to use it and it's uncommon in Ireland for the same reason you don't find many British girls named Brittania. Plus there is Erin Soup.