Question

what was the name of the mystic and healer who gained power by befriending the royal family?

Explanation:

This figure is most commonly referenced as Rasputin, a Russian mystic and self-proclaimed holy man who befriended the Russian imperial Romanov family, gaining significant influence and power through his perceived ability to heal the hemophiliac heir Alexei Nikolaevich.

Answer:

Grigori Rasputin