My mother said it is sunnah that a girl child should have her fathers first name instead of last. She said it is a sunnah to have the first name. Is it written anywhere that it is sunnah? If so could you please describe it more. Thankyou.
Surnames, as we know them, weren't really a thing for Arabs during the time of the prophet. Rather, they were identified using a patronym e.g. "Muhammad ibn Abdullah" (Muhammad, son of Abdullah) or "Fatimah bint Muhammad" (Fatimah, daughter of Muhammad). This sort of naming was already a long-established cultural practice rather than any sort of sunnah introduced by the prophet.
So it is definitely the case that the prophet and those who followed him did name themselves after their father's first name, but for the most part that's just because the first name was also the only name. The fact that this practice was continued by the prophet indicates that it is permissible, but doesn't necessarily suggest any sort of obligation or that it has any sort of blessing attached to it. It is simply how things were done at the time.
The only actual Islamic requirement in this issue that I know of is expressed in the Qur'an:
…ادعوهم لآبائهم هو أقسط عند الله
Call them by [the names of] their fathers; it is more just in the sight of Allah…
This requirement refers quite particularly to adopted sons and speaks against the common practice of the time to name an adoptee after the adopted father rather than their original birth father. Even if you generalize it to refer to all children, it really only specifies that they need to be identified as children of their birth father, not necessarily that their name needs to explicitly reflect that, and especially not that it needs to be on a first-name basis.