I'm more familiar with later Islam, and while I can't fully answer it, I think I can provide you with a couple pieces of important information for your question.

Firstly, there is one major piece of Arabian paganism that became a centerpiece of Islamic worship, the Ka'ba. Originally a polytheistic shrine to pagan gods, the Ka'ba was both an important holy site and a key source of revenue for the town from pilgrimage. After Muhammad took the city of Mecca in 630, the monument was cleansed of paganism and converted to be a focal point of the monotheistic worship of Allah. As you may know, travelling to the holy cities, including a visit to the Ka'ba is part of the Hajj, something all Muslims are expected to do at least once in their life, and remains an important source of both power and money to the Saudi Arabian government to this day.

Secondly, Fred Donner, in a lecture entitled How Ecumenical was Early Islam, argues that after the "definitive crystallization" of a new religion by said religion's intellectual leadership, many new followers still retain close ties to their previous community. This leads to the intellectual leaders attempting to "forcefully separate" from the religious confession that it arose from. Donner is using this to speak about seperation from earlier Judeo-Christian scripture, as though Islam arose in a primarily pagan area, it's doctrinal lineage is more closely tied to the Jewish and Christian ahl al-kitab, "the peoples of the book". However, I would argue that the cleansing of the Ka'ba demonstrates that similar pressures to distance the new faith from established beliefs were also felt by early Islamic leaders in regard to paganism. Furthermore, the Quran, like many religious texts, can be contradictory. It contains passages that are highly ecumenical towards ahl al-kitab as well as passages that are explicitly anti-Jewish and anti-Christian. Explanations for this discrepancy range from they represent Muhammad's changing views at different stages of his life to the idea that the Quran may represent an amalgamation of texts from different places in the Arabian peninsula and as such, represents the complex and differing views of various groups towards their neighbors. This could suggest that integration of these religious communities, and by proxy, their religious beliefs, was a contentious issue.

In conclusion, there is certainly evidence for the integration of pagan beliefs into early Islam. It also seems clear that Islamic intellectual leaders felt great pressure to distance themselves from earlier religious movements.

Source:

Donner, Fred M. "How Ecumenical Was Early Islam?" University of Washington Near East Languages and Civilization: Ziadeh Lecture 2013. Apr. 2013.

In general, it is fairly difficult to establish the form of pre-Islamic religion in Arabia outside of its description in Muslim sources, leading to a general difficulty of interpretation given the way that such sources were created by adherents of a "new" religion and thus inclined towards emphasising difference with previous forms (sometimes through misrepresentation of those forms). This is changing with recent archeological discoveries in the Levant and peninsula but there remains is a lot of uncertainty.

The first thing that springs to mind as having a clear continuity is the practice of Arabic poetry represented by the Qur'an. The Qur'an, believed by Muslims to be God's speech, uses and invokes pre-Islamic linguistic and poetic conventions. Its adherence to, and selective disregard, of those forms remains an important part of some approaches to Qur'anic exegesis. In some Muslim scholarly traditions one element of training in the interpretation of the Qur'an is the study (and sometimes memorisation) of pre-Islamic poetic collections like the "Mu'allaqat".

The structure of Arabic gives a lot of space for very poetic and metaphoric language, and the beauty of the poetry of the Qur'an is believed by Muslims to be miraculous: itself a proof of its divine source. The second verse of the Qur'an includes a direct challenge in relation to this belief:

And if you are in doubt concerning that which We have sent down (i.e. the Qur'an) to Our servant then produce a surah (chapter) of the like thereof and call your witnesses besides Allah, if you are truthful. (Muhsin Khan's translation of Qur'an 2:23)

The reason I would include this in the continuity of pre-Islamic practices is that the role of poets in pre-Islamic society was apparently not clearly distinguished from religious beliefs. Arabic poetry prior to the coming of Islam sometimes had elements of soothsaying, and in general poetry in its various forms apparently had an immense significance for pre-Islamic Arabs. While the coming of the Qur'an fundamentally changed both the Arabic language and many elements of Arabic-speakers' relationship to poetry, the ongoing import of poetic words (both the Qur'an itself and, later, religiously inspired poetry) for Muslim religious practice is arguably a continuity of its regard in pre-Islamic culture.

The challenge in the Qur'an to produce another of its like is also believed by some to be a direct continuation of a practice of poetic "battles" between tribes and their poets, whereby as a substitute for warfare their skilled orators would exchange boasts and jibes in a sort of pre-Islamic rap battle. It has not been, I believe, definitively established that this happened in pre-Islamic Arabian society, but it was something that happened in later eras.

The importance of pilgrimage (and related prohibitions around warfare) arguably continue over from pre-Islamic times, as u/Dur-eDurran relates in relation to the city of Mecca and the Ka'ba.

Belief in al- 'ayn, the evil eye, pre-dates Islam, as does the making and use of talismans to ward against it. This is something that continues in Muslim cultures and is at least partly addressed by early Muslim religious texts.

According to Muslim accounts of Islamic history, belief in a monotheistic God, Allah (from the Arabic al-ilah which means "the God") was something that existed in pre-Islamic Arabia. At the time of the coming of Islam, according to Muslim sources, some people would worship a range of dieties with Allah as the supreme diety, while others would worship Allah exclusively. In this respect the worship of a monotheistic God, devoid of anthropomorphism, was a pre-Islamic practice that was spread and formalised through both the efforts of a Messenger and the revelation of Allah's word. This is not an uncontested assertion given the difficulty of establishing the exact nature of belief in Allah outside of Islamic sources.

Such sources will often present the Prophet Mohammed as being faithful not just to the religion revealed to him, but also - through the early years of his life - to the monotheistic religion that existed prior to revelation. His seclusion in the cave of hira (where revelation is said to have come to him) is understood as having been a pre-Islamic religious practice (seeking seclusion from the world).

There is a lot more here that can be elaborated by others, especially the role of Djinn and other supernatural beings, but that's not something I know a lot about. This also mostly focuses on the Arabian peninsula, as the situation in North Africa, the Horn of Africa, and Central Asia I am also less familiar with.

Edit(adding some further stuff about Iran, if you forgive me including Zoroastrianism in the category of paganism, where it doesn't belong):

In the case of Iran there are several examples I can think of of the continuity of pre-Islamic practices, but the most obvious is the calendar. The festival of Nowruz, Iranian New Year, still celebrated in Iran today, has its origins in Zoroastrianism, following an astral calendar.

Henrietta Stepaniants, in an article titled "The Encounter of Zoroastrianism with Islam" argues that some Sufi texts show clear Zoroastrian influence. Given that Iranian scholarship and scholars were so important to much of the evolution of Islamic law and theology it is difficult to imagine that the influence of Iranian Zoroastrian thought was marginal or erased by the coming of Islam, and if anything may have spread beyond Iran through scholarly networks.

Aren't Jinn part of pre Islamic beliefs?