In his time, names in Italy were often differently structured than today, and they varried a lot. For example Leonardo's full name was Leonardo di ser Piero da Vinci ("Leonardo, son of ser Piero from Vinci"). For Michelangelo it was Michelangelo di Lodovico Buonarroti Simoni. For Raphael it was Raffaello Sanzio da Urbino. They all are known and commonly referred to by their baptismal name (often translated it. Raffaello => engl. Raphael, germ. Raffael).

However, there are other Italians contemporary to them, were we use other parts of their name as a shorthand, sometimes altered somewhat to adjust to our modern forename-surname system. For example: Filippo di ser Brunellesco di Lippo Lapi => Filippo Brunelleschi, or Niccolò di Bernardo dei Machiavelli => Niccolò Machiavelli.

There are other special cases: Giovanni Pico dei conti della Mirandola e della Concordia is known as "Giovanni Pico della Mirandola" and can be reffered to as "Pico della Mirandola" or "Pico", but usally not as "della Mriandola".

Galileo Galilei is also a special case, were both, "Galileo" and "Galilei", are acceptable (although I think "Galilei" is becoming more and more the standard).

So there is no rule of thumb how to refer to famous Italians of the Renaissance. It differs from case to case. You simply have to follow the individual practices.