The root would be 'madadh' (dental fricative) plus -án

The 'dh' then become 'gh' later, and might have been strengthened to 'g' (maybe An Lon Dubh knows) or it is due to English admin writing down the g?

MacLysaght says for Madden: Ó Madáin from earlier Ó Madadháin, Uí Maine sept, East Galway. A branch settled in Kildare

O Madagáin -branch of the O'Maddens settled in Clare and Limerick

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In earlier centuries, when 'dh' becomes 'gh' or just 'g', is that an organic, gradual change in pronunciation or just a scribal/spelling change? I'm working with a 19th-c. manuscript that replaces 'dh' with 'g' seemingly at random, but I can only assume this reflects the contemporary pronunciation as much as anything else.

Sounds like an Anglicization.

Redwolf

There is also fortification/de-lenition that sometimes happens ('cuimhneamh'/ 'cuimneadh' etc), maybe someone knows

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gh/dh is pronounced [ɣ​] only word initially in Modern Irish. Within words or at the end of words it has either changed or vanished.
-agán is used very often as a suffix on its own, so an analogous change from Madadhán to Madagán isn't very surprising.

So Madadhain IS pronounced with a G sound? Madagain?

You've got a spelling error there (it's Madadháin) but yes...to a point.

It's not a "g" sound so much as it is a "gh" sound. As if you started to pronounce a hard "g" but didn't close your throat all the way, letting some air pass through (Maybe someone can make a sound file for you).

Also that long "á" in the final syllable makes a big difference. It would be something like MAD-uh-ghawn.

Redwolf

Ó Madadháin is nearest 'O' Madden' in English, as it went through the steps of 'dh' then 'gh' then [zero]

The use of -agán is a suffix must have happened at a more recent date. 'O'Madigan' in English

Both can be considered the same name ultimately

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