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The prophet came to the world in a society that used to believe in many fake gods, and especially not in the great God.
The Prophet's father name is Abdu Allah which the translation to English is slave of The God. Note: Al_lah the Al in Arabic means is unique. So the exact translation would be slave of the unique God
My question is: How come the prophet's father name is slave of The God, while God was unknown or at least not recognized back then?
As goldPseudo explained in his answer, the name Allah was known to them then. Furthermore your question assumes that they did not even know about this one Unique God, that is not totally correct, they knew about Him, and they thought that worshiping the idols would get them closer to Him, but Islam came and said no this is not the Proper way and is Shirk. See Surat Azumar Ayah 3.
The word 'Allah' as a God was well known at the time of the pagan Arabs (and perhaps even more backward in the history).
In fact they even believed Allah to be the creator and sustainer:
If indeed thou ask them who has created the heavens and the earth and subjected the sun and the moon (to his Law), they will certainly reply, “Allah”. [...] And if indeed thou ask them who it is that sends down rain from the sky, and gives life therewith to the earth after its death, they will certainly reply, “Allah!” (Surah Ankabut, verses 61-63)
They also gave a lip service to the fact that Allah has no partners, by Chanting:
Labbaika Allahumma Labbaik, Labbaika la sharika Laka labbaik(you have no Sharik/partner O' Allah)
, but after uttering the above statement they added something like: "except you who gave {Idol name} power to be a partner"
Moreover, there existed a community among them which was called as 'Hanif' , and they were sincere monotheists and did not worship others alongwith Allah
What made the pagan Arabs Mushriks(One who commits polytheism) was that they worshipped other Gods along with worshiping Allah
And most of them believe in Allah, but commit shirk. (Surah Yusuf, verse 106)
Ibn Abbas (رضي الله عنه), when asked about this verse, said “their eman in Allah is that if you ask, ‘who created you?’ they say ‘Allah’. Their shirk is that they worship other then him.”
The excuse of the jahili Arabs to commit shirk is preserved in Surah Yunus:
They serve, besides Allah, things that hurt them not nor profit them, and they say: “These are our intercessors with Allah.” (Surah Yunus, verse 18)
These pagan Arabs believed their Idols(which included dead people, angels, stars, prophets etc) to be intercessors with ALLAH which bring them closer to ALLAH.
Thus it is believed by many groups like the Shia that the Prophet's father and his grandfathers were not pagan and that they belonged to this community of "Hanif"(Monotheists) who were always monotheists to begin with and so was the Prophet pbuh. Hence there is a possibility that the name of the Prophets father was well knowingly given to him in true spirit of monotheism.
Islam did not invent or introduce the name "Allah," rather the name was known to the Arabs beforehand. Islam merely denounced the pre-Islamic understanding of Him as shirk.
Before the advent of the prophet, "Allah" was just considered one of the many gods, with other gods and even his own daughters sharing similar roles.
