How did medieval Christian monasteries decide which Greco-Roman classics to preserve?

Answer 1

Generally those texts and authors which could be reconciled with Christianity were preserved and transcribed for future generations.

If you are wondering how an author like Ovid came to be spared, it's largely in part because medieval Christians interpreted texts like the Ars Amatoria and Remedia Amoris to be didactic and moralizing.

Occasionally non-Christian authors like Virgil also survived because they were deemed to be "pious pagans," i.e. even though they lived before the advent of Christianity or repudiated the religion, they were still seen to live moral lives.

Taste also goes a long way. Whatever was in vogue during late antiquity (roughly the 4th and 5th centuries AD) found champions in medieval monasteries.