What does the name of Allah mean in the eyes of the Arabs before Islam? (Given that the name of the father of the Prophet of Islam was Abdullah)
The pagans of Mecca believed in "Allah" as the head of their pantheon and the creator god, with other deities ascribed as his children.
Islamic history states this was because the Arabs were originally monotheists, being the descendants of Abraham through Ishmael and followed Abraham's religion of aniconic monotheism ... though over the ages corruptions such as polytheism and idolatry crept in.
Abraham and Ishmael built the Kaabah in Mecca and Ishmael and his progeny settled there, and the Quraysh and other Arabian tribes claimed descent from him.
Quran 2:127-129 And [mention] when Abraham was raising the foundations of the House and [with him] Ishmael, [saying], "Our Lord, accept [this] from us. Indeed You are the Hearing, the Knowing. Our Lord, and make us Muslims [in submission] to You and from our descendants a Muslim nation [in submission] to You. And show us our rites and accept our repentance. Indeed, You are the Accepting of repentance, the Merciful. Our Lord, and send among them a messenger from themselves who will recite to them Your verses and teach them the Book and wisdom and purify them. Indeed, You are the Exalted in Might, the Wise."
Quran 14:35-37 And [mention, O Muhammad], when Abraham said, "My Lord, make this city [Makkah] secure and keep me and my sons away from worshipping idols. My Lord, indeed they have led astray many among the people. So whoever follows me - then he is of me; and whoever disobeys me - indeed, You are [yet] Forgiving and Merciful. Our Lord, I have settled some of my descendants in an uncultivated valley near Your sacred House, our Lord, that they may establish prayer. So make hearts among the people incline toward them and provide for them from the fruits that they might be grateful.
قال رسول الله صلى الله عليه وسلم " ان الله اصطفى من ولد ابراهيم اسماعيل واصطفى من ولد اسماعيل بني كنانة واصطفى من بني كنانة قريشا واصطفى من قريش بني هاشم واصطفاني من بني هاشم
The Messenger of Allah said: "Indeed Allah has chosen Ishmael from the children of Abraham, and He chose Banu Kinanah from the children of Ishmael, and He chose the Quraish from Banu Kinanah, and He chose Banu Hashim from Quraish, and He chose me from Banu Hashim."
You can find a detailed accounts of this in e.g the Seerah of Ibn Kathir here.
Regarding the origin of the name Allah. Its a special form of ilah (god) and is cognate to the Hebrew Elohim which is from Eloah (also see Aramaic Elah) which is thought to be derived from the semitic name EL (as in GabriEL and IshmaEL), all forms occur in the Torah\Tanakh to refer to God.
Examples of their usage in the Tanakh\Old Testament:
Bereshit, bara elohim, et hashamayim, ve'et ha'aretz
In the beginning God created the heaven and the earth.
Vayishman yeshurun vayiv'at, shamanta avita kasita; vayittosh eloah asahu, vaynabbel tzur yeshu'ato
And Jeshurun became fat and rebelled; you grew fat, thick and rotund; [Israel] forsook the God Who made them, and spurned the [Mighty] Rock of their salvation.
bishnat ẖada lekhovresh malkaˈ kovresh malkaˈ sam teˈem bet-ˈelaha virushelem
In the first year of Cyrus the king the same Cyrus the king made a decree concerning the house of God at Jerusalem.
— Ezra 6:3
Arabs knew Allah before Islam. Their problem was that they didn't worship Him alone.
Quran explains this in many places, for example:
And if you should ask them, "Who has created the heavens and the earth?" they would surely say, "They were created by the Exalted in Might, the Knowing." 43-9
Unquestionably, for Allah is the pure religion. And those who take protectors besides Him [say], "We only worship them that they may bring us nearer to Allah in position." Indeed, Allah will judge between them concerning that over which they differ. Indeed, Allah does not guide he who is a liar and [confirmed] disbeliever. 39-3
As you mentioned also, Arabs before Islam had names like Abdu-Allah, (Allah's servant).