I'd have two partially worked out designs - one for ancient Persia, other for islamic Persia. For each I have two leaders, each from different dynasty. Gimmicks for unique units and buildings I was too lazy to think of, and I outright admit I wasn't very original when thinking of unique buildings either

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Persia (Ancient)
Unique ability:
Satrapies - in conquered cities, Persia may recruit unique unit of the civilisation that founded said city, provided said civilisation had unlocked said unit. I suppose something regarding stability of these conquered cities could work, too.
Unique unit:
Immortal
Unique building:
Pairidaeza and/or
Caravansarai
Achaemenid leader:
Darius I
Capital:
Persepolis or
Susa
Leader ability:
Royal Road - In times of war, Persian military units move with extra speed on roads built inside Persian territory. In times of peace, non-military units move with extra speed on these roads.
Note: One of the most important functions of the Royal Road Darius I had built during his reign was it serving as a highway for messengers. Creating a very effective postal system for his time, Darius could receive messages from the western border of the Achaemenid Empire in his royal palace in Susa in less than two weeks' time - an astonishing speed for the time. Needless to say, it was needed - the western border got quite busy under his reign, after all

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Sassanid leader:
Khosrow I
Capital:
Ctesiphon
Leader ability:
Wek Antiok Khosrow
Note: Khosrow I was a powerful ruler enough to become a feared rival of the Eastern Roman Emperor Justinian, not only engaging him in wars, but also securing victories against the Byzantines, all while successfully expanding the Empire in other directions and simultanously bringing it internal stability. A decent choice for a Sassanid ruler, though I haven't much idea what skill to give him, so I went for a placeholder. During his wars against the Byzantines, Khosrow sacked the city of Antioch, and a few other surrounding towns in Byzantine Levant, capturing the locals. Those locals he had transported to live in a city inside the Sassanid borders he ordered to be built for them. The city was named Wek Antiok Khosrow. Meaning of the name? "Better than Antioch is this city built by Khosrow." Talk about adding insult to injury

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Persia (Islamic)
Unique ability:
Lion and Sun - Frankly, no idea. Culture bonus would be nice, though I have a hard time thinking of something distinctively Persian at the moment

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Unique unit:
Zamburak
Unique building:
Pairidaeza and/or
Caravansarai
Safavid leader:
Abbas the Great
Capital:
Isfahan
Leader ability:
New Imperial Capital - Unlocks a project in all other cities that, once finished, transports 1 population from the city to the current capital of Persia. All structures providing housing have their housing yields increased in the Persian capital. Districts and buildings in the capital are 25% cheaper to erect, and the capital receives 20% speed towards wonder building.
Note: Probably the most accomplished Safavid Shahanshah, Abbas the Great did many things, such as massive military expansion of the empire, strenghtening of Persian economy, achieving the golden age of Safavid culture or establishing foreign relations between Persia and European powers, but as a Persian architecture enjoyer, I went with the Shah's project of moving the capital from the city of Qazvin to Isfahan, a town further away from dangerous Ottoman foe. In the process of establishing of the new imperial capital, Abbas transformed the city into one of Asia's most thriving metropolises of the time, complete with a grand set of magnificent projects he ordered built in the city - Naqsh-e Jahan, Ali Qapu or Shah's Mosque.
Afsharid leader:
Nadir Shah
Capital:
Mashad
Leader ability:
Sack of Delhi - When pillaging a structure housing a great work and Persia has any free space for corresponding great work in one of its cities, said great work gets transferred to Persia.
Note: During his reign over the Afsharid Empire, Nadir Shah invaded the Mughal Empire in India, heavily crushing it militarily before entering Delhi, the Mughal capital, sacking it. The Persian Shahanshah captured immense wealth from the city (the seized wealth was so great that he temporarily halted taxation in Persia after his return) and managed to collect very great amount of treasures which he ordered transported to Persia - the most notable among those the famous Koh-i-Noor diamond and the Peacock Throne. The two mentioned items may not be in Iran anymore - the Peacock Throne was lost to history after Nadir Shah's assassination and Koh-i-Noor is now in British possession, but still, till this day, parts of the collected treasures from Delhi are to be seen in the Iranian capital of Tehran.
Side note - as anachronistic as it might be since it was used in 18th and 19th century mostly by the Qajars, I'd absolutely love if the Persian palace Islamic Persians would use was based on the
Golestan Palace, or at least if the Golestan Palace was added as a wonder.