The village of Atherton in Greater Manchester (formerly, in Lancashire) has recently been in the news. It is quite near where I grew up, so I am confident in saying that it is pronounced ˈæðətən.
As we know, the spelling digraph th regularly corresponds to two distinct phonemes in English: θ as in thin and ð as in this. (For the moment we can forget the occasional irregular correspondences, as for example to t in Thomas.)
The rule is easy in word-initial position. In LPD I expressed it like this:
Word-medially, it generally depends on whether or not the word is of Germanic origin:
Since Atherton is obviously of Germanic origin (‘farmstead of a man called Æthelhere’), it is indeed expected that the fricative would be voiced, as also in Brotherton, Netherfield, Rotherham, etc.
As a surname, however, Atherton is often pronounced with a voiceless fricative. I have to wonder how the places of this name in California, Indiana, Ontario and Queensland are pronounced.
Even more surprisingly, Atherstone in Warwickshire, according to the BBC Pron Dict of British Names, has θ (though Wikipedia says it has ð). So does Athelney in Somerset. Athelstaneford in Scotland is a law unto itself, being ˈaθl̩stenfɔrd or even ˈɛlʃənfərd. Scottish Atholl is ˈæθl̩. And the Athenry whose fields are commemorated in song by Irish nationalists is ˌæθənˈraɪ; but then the origin of this name is not Germanic but Celtic (Irish Átha an Rí ‘the king’s ford’).