I've got this weird theory that Monet is Hattori's daughter from the future. She was sent back in time to undo the mistakes of the World Government, which is at fault for destroying all of civilization in the near future, but fell in love with Doflamingo and decided to not follow her orders, and thus the time loop remains closed.
She eventually leaves Doflamingo because she realizes that he doesn't give a shit about his subordinates, no matter how close she gets with him. However, Doflamingo, belying his appearance and manners, was no easy-going man. He hunted her down and cornered her on a remote sacred island of the birds. But, chasing her to this island proved to be a catastrophic mistake for Doflamingo, as it's beneficent dictator Hattori felt compassion and affection for the young bird-maiden, and shielded her from harm with a gust of wind that blew away him and his strings. Unfortunately for Doflamingo, those razor-sharp strings rebounded towards his neck and limbs from the forces of the winds and effectively dismembered him as he plummeted into the seas.
Ingesting the fruits of the islands flora, Monet eventually transforms fully into a pigeon. In a heat of passionate courtship, Hattori took Monet to be his betrothed.
Monet bore a single egg, which is odd for a pigeon, but it reflected her human origins. That egg hatched a little girl, and was named Monet after her mother.
Monet was a rambunctious young sprite. She would often board random boats, innocent of the nature of pirates, marines, or merchants, but her agility allowed her to escape capture.
Her mischief developed the skills to become an infamous thief. And despite her skills and her heritage, she was caught by the Marines and imprisoned.
That act infuriated the mighty Bird Lord. Declaring war on the World Government, Hattori and his armies decimated the human world. As the world crumbled from Hattori's fury, Vegapunk created a time machine and chose young Monet to be it's first voyager. She was sent back in time to prevent her own capture, but the prototype malfunctioned, instead sent her back to a time before she was born.
She became bored of waiting, since the time machine was a one-way gate, and decided to explore the world. She met a charismatic pirate lord named Doflamingo and was enlisted as his subordinate.
The rest, is history.