Is it permissible for someone to have their father's first name as their surname in Islam, instead of their father's surname as their surname?
It is permissible. Allah says:
ادْعُوهُمْ لِآبائِهِمْ هُوَ أَقْسَطُ عِنْدَ اللَّهِ
“Call them by their fathers’ (names), it is more just with Allaah” [33:5]
Example with first and last name: Ayesha bint(daughter of) Ahmed(Father name) or Ayesha Ahmed(you may remove bint to avoid confusion in the travel documents.)
Example with first, middle and last name: Ayesha(First) Ahmed(Father) Khan(Last name/family name)