Which evolved with the passage of time to become its Italian equivalent, Medici.
So the reason that Lorenzo De Medici becomes Leonardo DaVinci's patron is that the family vaguely remembers The Doctor and wants to honour him by sponsering the arts and sciences. Not sure where you would go with this, but I'm sure that someone can probably link it in to the various episodes that reference the Medicis, Leonardo, etc.
If anyone wants to use this please do, but a credit of some sort would be nice.