My understanding is that the influence of the Hagia Sophia on Islamic monumental architecture was profound. According to Andreas Kaplony, the earliest monumental architecture in Jerusalm was an explicit attempt to demonstrate that Byzantine architecture and building skills were now being employed by Muslim rulers for the benefit of their "Muslim Byzantine" subjects. And if I recall from architectural history correctly, Hagia Sophia was the explicit model for the domed Islamic mosque, where the dome came to be associated with the vault of heaven.

Interestingly, if you visit or view photos of the Hagia Sophia, you'll see that there is a kind of a "kludge" on the sides. The end apses are topped by half-domes, which help distribute the lateral forces from the weight of the domes more evenly to the ground. The sides, however, have massive buttresses on them that serve the same function of the later Gothic flying buttresses. Over (a long!) time, this has caused some instability, as is apparent if you walk the building now. The Islamic model, by contrast, was to put semi-domes on all four sides, as in the adjacent Blue Mosque.

Byzantine architectue also had a profound influence on the Church of the Holy Sepulcher btw, though it may be hard to recognize now. The original supposed spot of Jesus' burial was transformed into a circular memorial with a Roman-styled (the Byzantines were the evolved Roman Empire, remember, and they called themselves Romans) entry sequence. The circular memorial, called the Anastasius, was styled after a typical Byzantine memorial that marked the graves of important leaders or important spots. I think some historians will argue that the Dome of the Rock was built in part to compete with the Church's dome and to redirect attention to the supposed spot of the Temple. In that reading, both the early Christian and the early Muslim rulers were using the prestige of Byzantine architecture to apply to their own messages.

Hagia Sophia specifically became a model for Ottoman mosques after Ottoman rule came to Anatolia in the 16th century—at which point Hagia Sophia itself became a mosque. Mimar Sinan, the chief architect for sultans Suleiman the Magnificent, Selim II, and Murad III, was responsible for hundreds of iconic buildings including many mosques that were influenced by Hagia Sophia—namely in the incorporation of a soaring dome. Generations of Ottoman architects followed Sinan’s model, and since the Ottoman empire once controlled large parts of the Muslim world, this aesthetic spread widely. If you are thinking of domed mosques that resemble Hagia Sophia, you are probably thinking of buildings dating to the Ottoman era or later, and have Ottoman architects like Sinan to thank. (It’s also worth noting that Hagia Sophia has many Islamic era modifications, like the minarets.)

That said, if we go back further, Byzantine and Roman architecture has a much deeper influence on mosques. First of all, many areas that came to be part of the Muslim world were once Roman provinces, and so Roman architecture was part of the cultural heritage of much of Arabia, North Africa, and the middle east; naturally architects would look to the architecture around them for inspiration, including things like the basilica plan, but also decorative elements like stucco work, mosaics, etc. Over time buildings once used as Roman era pagan temples often also became churches and then mosques, as power shifted in specific localities. If we’re talking Byzantine specifically, Byzantine architects were commonly hired to work on Islamic architecture. Buildings like Dome of the Rock show a deep Byzantine influence, for example, both in the domed plan of the building and in the mosaic interior decoration.

The Umayyad Mosque of Damascus, one of the earliest monumental mosques, is a great example of all of this. It was built on a site that had been used as a place of worship for many centuries already—in the Roman era there was a temple to Jupiter there, and then a Christian church. The building incorporates spolia from the Roman era including Roman columns, and follows a basilica plan. Byzantine architects and artisans were employed in its construction, and the mosaics again show a deep debt to Byzantine mosaic work.