In my understanding, your question mostly comes down to a few assumptions about the period within which Abdullah lived: namely, that "Allah" refers exclusively to the Islamic god, and that polytheists did not call any of the entities they venerated "God".
The most important thing to note here is that "Allah" does not exclusively refer to God as understood by Muslims; rather, it is simply the Arabic-language word for God, and indeed is the term of choice for modern Arab-speaking Christians, particularly the Maronite Catholics of Syria and Lebanon.^1 This is similar to the term God itself in English, which is used to describe deities in general but which, in certain contexts, can also refer to specific Abrahamic conceptions of God, including as understood by both Jews and Christians alike. As such, it's best to understand the term Allah as one which could refer to a variety of different deities, depending on the context.
Beyond that, however, the question still remains: why would a pagan man be named "Servant of God"? In this case, we have to look at the broader circumstances from which Islam arose. The Arabic tribes of the region were no strangers to Christianity, and certainly not strangers to the idea of a monotheistic god more generally: note that besides the Roman empire, the other major polity with which tribes in the area would have most often interacted, the Zoroastrian Sassanids, also believed in a supreme being, albeit one with a weaker, malevolent, dualistic counterpart (EDIT: see /u/lcnielsen 's comment below for some qualification on this note). Most notably, the tribe to whom Abdullah and Muhammad were born, the Quraysh, did have a conception of a god named "Allah", after whom Abdullah would thus have been named.^2
There's some debate around the exact details beyond this point, so I'll mostly be going off of what Scott Fitzgerald Johnson has to say: no one work can include every perspective, but I feel like he does a pretty good job here. Allah was likely one of: A) one god among many in the pantheon venerated by the Quraysh; B) another name for the head of the Qurayshi pantheon; C) the name of a creator god who had since fallen out of favor in the local myth. In any case, he eventually became rather more significant for the Qurayshi after his intervention was credited for their victory over an army assembled by a Christian Arab King to conquer Mecca. After this point, sources describe the Quraysh as having adopted the ascription "ahl Allah": Allah's People. We can thus assume that it was for this reason that from among all the gods venerated by the polytheistic Qurayshi, Abd al-Muttalib (the Prophet Muhammad's paternal grandfather) chose Allah in particular to venerate via his choice of name for his son.
Allah was already worshipped as the local god (or one of the local gods) of Mecca, and it was only after Muhammad that he came to be regarded as (or, from the Islamic perspective, was revealed to be) the sole God, the creator of the universe. So it wasn’t too unusual to be named the “servant of Allah” and there were several other people of Muhammad’s generation and the previous generation named Abd Allah.
The period before Islam is known as the jahiliyyah, the age of ignorance, when the tribes of Arabia were either pagans, or Christians or Jews. Pagan tribes and towns all had various local gods. Allah was one of the gods worshipped in Mecca:
“Allah was known to the pre-Islamic Arabs; he was one of the Meccan deities, possibly the supreme deity and certainly a creator-god…But the vague notion of supreme (not sole) divinity, which Allah seems to have connoted in Meccan religion, was to become both universal and transcendental; it was to be turned, by the Kur’anic preaching, into the affirmation of the Living God, the Exalted One.” (Gardet, pg. 406)
Mecca had a sanctuary, a haram, which was meant to be neutral ground where everyone from the surrounding area could meet peacefully, which was important for a tribal society where the different tribes were constantly raiding each other. This also made Mecca a significant commercial centre, since everyone could get together and trade without fear of being attacked. In the 6th century, the guardianship of the haram of Mecca was taken over by Qusayy of the Quraysh tribe. As guardians of the haram, the Quraysh were the “holy family” of Mecca and became quite wealthy and powerful. As it still does today, the haram contained the Ka’ba, which at the time housed various idols representing the various local gods, including Allah.
Qusayy’s great-grandson was Abd al-Muttalib, who had numerous children, including Abd Allah; hence he is also known as Abd Allah ibn Abd al-Muttalib. Not much is said about Abd Allah in early Muslim traditions, except that he died young, around the age of 25, either before Muhammad was born or shortly after his birth around 570. At the time, the Quraysh and other inhabitants of Mecca probably regarded Allah as their chief god among all the other gods represented in the Ka’ba. Allah was also believed to have consorts and children. The etymology of the word might actually be simply “the god”, containing the Arabic definite particle al- and ilah, a word derived from the root for “god” in Semitic languages (which also gives us El or Eloh in Hebrew, and Alaha in Aramaic).
With the revelation of the Qur’an to Muhammad, Allah’s name came to be understood not just as the main local god, but the one and only God, the creator of the universe. All other gods were then recognized as false idols. So after the advent of Islam, the name “Abd Allah” was much more significant, as it now meant a servant of the one God, not just a servant of the local god whose name happened to be Allah. Now, it is by far one of the most popular names for Muslims.
But there were several other Abd Allah’s in Muhammad’s time, who predated Islam. One of Muhammad’s opponents in Medina was named Abd Allah ibn Ubayy, and Muhammad had cousin named Abd Allah ibn Abbas (son of Abbas, one of his father’s brothers). One of Muhammad’s companions and early converts to Islam was named Abd Allah ibn Mas’ud. Two other notable early converts to Islam were Abd Allah ibn Salam and Abd Allah ibn Saba - and they were originally Jewish Arabs, so the name was apparently popular with the Jews as well (although presumably they interpreted “Allah” as referring to the one God of Judaism).
Sources:
Louis Gardet, “Allah” in Encyclopaedia of Islam, vol. 1, 2nd ed. (Brill, 1960)
Hugh Kennedy, The Prophet and the Age of the Caliphates: The Islamic Near East from the Sixth to Eleventh Century, 2nd ed. (Longman, 2004).