Anon, that is a dangerous topic to get me started on. I have quite a few (and I'm omitting some of the big names that I also adore: Arjumand Banu, Mihrunissa, Hamida Banu Begum, etc.)
Khanzada Begum- Babur's sister, and--depending on which version you believe--either abandoned or straight up sold to Babur's enemy Shaybani Khan during Babur's escape. But! she wound up surviving both her marriage to the enemy, and returned to her now-victorious brother, subsequently marrying again under happier circumstances and becoming (according to some sources) the first Padishah Begum* of the empire.
*I feel like people often mistranslate this title as "Empress" or "Emperor's favorite wife," but it could just as often mean a beloved sister or daughter--basically, it was the most important woman in the royal family, and the one who effectively claimed the most power.
Gulbadan Begum - half-sister of Humayan, historian, and straight up nerd. Wrote her brother's biography, and delighted in telling embarassing anecdotes about him like any proper sibling. Loving aunt to Akbar; clever, independent, and fun.
Mah Chuchak Begum - widow of Humayan, ruler of Kabul (initially in the name of her son, apparently later just gave up on the pretense and ruled by herself). A hurdle in the path of the young Akbar's consolidation of the empire, she was unfortunately killed by her son-in-law. That said, years later, after her son had managed to piss off Akbar, her daughter Bakht-un-Nissa Begum wound up inheriting the governship of Kabul and apparently did a bang-up job of it.
Aram Banu Begum- Akbar's younger and favorite daughter, and explicitly a smart-aleck. Apparently Did Not get along with her half-brother Jahangir, to the point that one of Akbar's dying wishes was that the two get along. Never married, but seems to have been more a personal choice, rather than a strict decree against it. Seems to have been A Lot, in the best of ways.
Nadira Banu Begum- wife of Dara Shikoh, arguably in one of the happiest marriages of the dynasty. Dara never married anyone, and like her mother-in-law before her, Nadira joined her husband in exile and revolution. He apparently gave most of the paintings he loved to create to her, which is adorable; and did not survive her death by more than a few months.
Dilras Banu Begum - wife of Aurangazeb; apparently haughty and beautiful, and not a little terrifying. Interestingly, she was a devout Shia while Aurangazeb was a devout Sunni (to be fair, Shah Jahan and Mumtaz Mahal were also a very happy Sunni/Shia pair, but both Dilras and Aurangazeb were famously pious and much more religiously fixated than the prior generation). Died before her husband became Emperor, but her children would succeed him.
Zeb-un-nissa (&her siblings) - daughters of Aurangazeb, and talented poets, scholars, and artists. Particularly notable because while pop cultures has the later Mughal empire portrayed as either a joyless husk, or a decadent waste -- these women were clear contraindications to that generalization.
(I could go on, but these are the first few that come to mind!)
