Manx bmd has a Mona Mitchell born 1902 in Douglas, entry 1629 I think, image a bit blurry. There is also a Christina Mitchell born 1903, possibly 1904 but probably 3, again poor image, born Douglas, entry 1722. The parents are not named on this record so you would have to apply to the registry for the birth certs to confirm the parentage. Cannot see a marriage for parents on Island or an entry in 1901 census. Did they move over just prior to Mona's birth? If you do not have Mona on 1911 census is there a possibility that she could be the Mona Mitchell buried Braddan cemetery aged 7mths, 8 October 1902? Perhaps some kind person on this board would be able to give you an MI.