Surnames O'Daly, Daly, Daley, Daily, Dailey, Dayley and Dawley,
and other forms of O'Dalaigh (pronounced O'Dhaulee)
The common definition of the O'Dalaigh surname today is, "deriving
from Dalach meaning 'one who is present at assemblies'; the root
word is Dail, now the official title of the parliament of the
Republic of Ireland". A connection is also possible to the long
tradition of scholarship and poetic achievement associated with
those who bear it, since the ollamh of Gaelic Ireland had a place of
honor at the tribal dail as a man of learning and a poet.
Other evidence points to an even older more significant meaning,
based on the claim by the pagan Irish that they were offspring of
their gods. This evidence is found on several Ogham stones which
contain the oldest known form of Irish writing.
An example from the Gowran Stone;

DALLO MAQA MUCORI MAQI ERACIAS MAQI
LI
and one from the Dunbell Stone of Kilkenny

BRANITTOS MAQI DECARI DDALLOS
These inscriptions appear to invoke either pagan gods or
mythological figures with names similar to the ancestral "Dalach". A
third ogham, Monataggert II further specifies the Dalach (Dalagni)
as sons of the eponymous ancestor (Dali). In many cases the
mythological ancestor was female.

DALAG N I MAQ I
DALI
From this evidence and other data associated with times of
antiquity in Ireland, it would appear that there are reasonable
grounds for assuming the family name "O'Dalaigh" has a godly or
mythological significance.Indeed, it provides a more logical
probable meaning of the name O'Dalaigh than the more popular
versions built almost entirely on definitions given in modern
dictionaries for supposed parts of the family name.