I don't have my father's last name. I want to get my name changed; I would like to know the ruling on changing of one's last name if your father is non-Muslim.

Praise be to Allah, the Lord of the Worlds; and blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions. If this will lead you to be affiliated with a genealogical tree that is not the one of your own father or of your own family, it is then forbidden as the Prophet Muhammad (Blessings and peace of Allah be upon him) said: "One of the greatest lie is to call a person after a man who is not his/her father". [al-Bukhari]. al-Bukhari and Muslim also narrated from Saad Ibn Abu Waqas and Abu Bakrata that the Prophet Muhammad (Blessings and peace of Allah be upon him) said: "Whoever is called after a man who is not his father and he knows that, so the paradise is forbidden for him". It is narrated in a sound Hadith from Ali that the Prophet Muhammad (Blessings and peace of Allah be upon him) said: "If anyone makes a false claim to paternity, or being a client of other than his own masters. There is upon him the curse of Allah, the angels, and all the people. Allah will not accept from him any recompense in the form of obligatory acts or supererogatory acts on the Day of Resurrection". [Narrated by Muslim]. As regards the fact that your father is non-Muslim, it doesn't affect you in any way in this life or in the Hereafter. You are not asked to change his name even if his name is among the names that are forbidden or dislikable in Islam. The fathers of most of the companions of the Prophet Muhammad (Blessings and peace of Allah be upon him) were not Muslims. The names of some of their fathers meant the devotion and slavery to idols or other things not to Allah the Almighty. More than that, some of the progenitors of the Prophet Muhammad (Blessings and peace of Allah be upon him) had such names and he didn't change their names, example 'Abdul Mutalib, Abdul'uza. Allah knows best.