There were certainly some figures of note from Islamic Palestine during the early Islamic period. One of whom is al- S̲h̲āfiʿī, the namesake of the S̲h̲āfiʿī mad̲h̲hab (school) of Islamic jurisprudence, which grew to be one of the four main schools of Islamic law. While his origins are disputed, the scholarly view of his birthplace is summarized quite succinctly by the Encyclopedia of Islam article on his life:

According to the most ancient preserved source (Ibn Abī Ḥātim al-Rāzī (d. 327/939),* Ādā**b al-S̲h̲āfiʿī wa-manāḳibuhu , Aleppo n.d., 21-3), al-S̲h̲āfi*ʿī was born either at ʿAsḳalān, a town on the southern coast of Palestine, or in the Yemen, while most biographers incline rather towards G̲h̲azza, likewise in southern Palestine (also mentioned, less frequently, is Minā near Mecca).

One of those most fascinating figures of the early Islamic period is the 10th century geographer al- Muḳaddasī, who hailed from Jerusalem (his name is based on one of the Arabic names for Jerusalem, Bayt al-Maḳdis). Muḳaddasī's most famous work is his "Description of Palestine and Syria", considered "the best representative of Arabic geography in the second half of the 4th/10th century." Muḳaddasī provides us with a wide ranging description of the Islamic world, which includes vivid descriptions of the Palestinian towns and villages of the period. One of his most widely cited accounts is of the Al - Aqsa Mosque and the origins of the term Sham (often translated as Syria or the Levant), which I will provide below:

Al- Muḳaddasī's description of the Al - Aqsa Mosque is especially noteworthy because it explicitly connects Jerusalem and the mosque with Muhammad's night journey, and further because it describes the nature of the mosque's dome. His comments on the Al - Aqsa Mosque are as follows:

In this Mosque is the Mihrab of the Khalif 'Omar... On the roof I read the following inscription : In the name of Allah the Compassionate, the Merciful. Praise to him who brought his servant {Muhammad) by night from the Masjid al Haram {at Makkah) to the Masjid al Aksa (at Jerusalem) on the precincts of which we invoke a blessing. May Allah give aid to his servant and vicar., 'Ali Abu-l Hasan adh Dhahir-li- Izazi-din Allah, the Commander of the Faithful. Allah's benediction be upon him and upon his immaculate forefathers, and upon his beneficent sons. For the restoration of this same Dome and its gilding, hath given command our illustrious and dear lord, the chosen servant of the Commander of the Faithful, and his devoted servant, Abu-l Kasim 'AH ibn Ahmad—Allah give him aid and protection !

His comments on the etymology of the term Sham are as follows:

It has been said that Syria is called 'Sham,' says Mukaddasi, " because it lies on the left of the Ka'abah, and also because those who journey thither (from the Hijjaz) bear to the left or north ; or else it may be because there are in Syria so many Beauty-spots, such as we call Shamat—red, white and black—(which are the fields and gardens held to resemble the moles on a beauty's face).

Of course, these are only excerpts, and al- Muḳaddasī's description of early Islamic Palestine is wide ranging (the original text is difficult to come by in translation, although Guy Le Strange's 19th century, "Palestine under the Moslems" quotes it often.)

To conclude, while the number of important Arab figures from Palestine during the early Islamic period were not as high as those hailing from Iraq or Persia, there were two very influential and important figures, whose works and deeds I have just described. If you are interested in military and political figures from this period, I would recommend "A History of Palestine: 634 - 1099", authored by Moshe Gil.

Sources:

Encyclopedia of Islam - al- S̲h̲āfiʿī, authored by E. Chaumonth

Encyclopedia of Islam - al- Muḳaddasī, authored by, A. Miquel

Palestine under the Moslems - authored by Guy Le Strange.