Hi iasha
We seem to be related, as Sophia Jane's sister Fanny is my 3xgreat grandmother. There is a lot of correspondence on the this site on Ballard/Goddard. The most comprehensive is under the topic 'Hindon Baptisms and Burials for PERRETT' started by shaunadavid.
The reason that the Ballard and Goddard names appear is because most members of the family appear at different times with either name. The best illustrations of this are Mary the mother who had both names on her burial entry in the parish register, and Fanny who is referred to as both Fanny Ballard and Fanny Goddard on her marriage certificate.
I have not tracked down our 4xgreat grandparents John and Mary Ballard. They seem to have originated from Hampshire and had the children Richard and Elizabeth born West Meon, Ann born Titchfield and Louisa born Kilmeston
before arriving in Winterslow. All of these were baptized as Goddard with parents John and Mary Goddard. {I originally was convinced that the family came from Ringwood but, with prompting from Valda, I have changed my view on this}
The family changed its name to Ballard when it arrived in Winterslow. So Sophia Jane who was baptized 15 June 1817 was baptized Ballard with parents John and Mary Ballard, and Fanny was baptized Ballard 18 Mar 1821. However, Sophia Jane was married as Goddard with father John Goddard on 1st June 1844 at Winterslow, whereas Fanny was married as Fanny Ballard akas Fanny Goddard with father John Ballard.
Burials were John Ballard 22 Aug 1843 aged 73, Mary Goddard alias Ballard 12 Feb 1866 aged 90 and Sophia Jane 9 Dec 1887 aged 71. All lived in Gunville. This is at the eastern end of the village and runs from The Common up to the Shripple. The men were mainly labourers and many were described as coppice labourers. This suggests that John Ballard and his son Richard, who changed his name from Goddard to Ballard, worked in the woods.
I hope you find this useful. [Suggestions from anybody as to why the family name changed between Goddard and Ballard will be most welcome.]
Take care
Colin