There are several ways the family's surname could have changed from the original Italian one which probably won't make it any easier to trace.
1) by sound- changed to something that sounded like, or a bit like, the original
2) by spelling- changed to something that looked a bit like the original
3) from something in the new country, a place or another surname. For example, my cousin's mother was a Kelly whose family came from Ireland and settled in Bronx, New York City. Lots of other Irish Kellys there but apparently there was one Chinese Kelly family there also. When they arrived from China they wanted 'an American' name and thought since there were so many Kellys it must be an American name.
4) surname chosen for them by a UK official- probably even more likely if they were illiterate or semi-illiterate
However, it may be possible to find the name based on the following-
She thinks it was Dora’s Father Antonio/Agostino/? who made the name change for some reason.
From this it sounds as though Dora's parents could have moved to the U.K. (hopefully Scotland) so would be worth searching for them in census records to start with. Father's Christian name possibly Agostino/August/Angelino/Angelo and mother Maria. It may be that they joined relatives already in Scotland or that others from their home in Italy followed.
