Tadhgan is spelt Tadhgan, and it's a boys name, a diminutive form of
Tadhg (teeg).
Teagan arises from confusion. There are two welsh names,
Tegan (teg-ann) and
Megan (meg-ann).
Megan was mistaken by
Colleen McCullough for an Irish name (the Thorn Birds). The spelling
Meghan developed. The name name became confused with the Irish surname
Meaghan (mee-un), and an evil hybrid,
Meaghan (meeg-un) arose (as the Irish name shows, the h should make the g silent. Doh!). So when people came across
Tegan in name books, they assumed it was tee-gan (possibly vaguely aware of an Irish name
Tadhg - see that h making the d silent?) and respelt it as
Teagan, thus perpetuating the whole horrible pile of ignorance.
It would probably have made things simpler if the Celtic languages had chosen another alphabet - this would cure people of the delusion that they understand the qualities of the letters they are looking at, just because English is written in an alphabet that looks the same.
Oh whoops. Did I say all that out loud?