Learn Italian: Un restauro minuzioso richiede pazienza, e la comincio ad avere. - A meticulous restoration requires patience, and I am starting to have it.

This is an example of clitic climbing, which is an option you have with aspectual verbs like cominciare a (to start to). You can either put the pronoun before the conjugated verb, as in la comincio ad avere, or attach it to the infinitive: comincio ad averla. Both are completely natural and mean the exact same thing.

What exactly does the la refer to in this sentence?

It refers back to pazienza (patience). Since pazienza is a feminine singular noun, it is replaced by the feminine singular direct object pronoun la ("it").

Why is there a d in ad avere? Why not just a avere?

The verb structure is cominciare a, but when the preposition a sits right next to a word starting with the same vowel (like avere), Italian often adds a d to make it sound smoother. This avoids an awkward double "a" sound.

Is the word minuzioso related to time, like "minutes"?

It is related to the English adjective "minute" (pronounced my-NOOT), meaning very small or detailed. In Italian, minuzioso means meticulous, thorough, or highly detailed, which is exactly what a good restoration needs to be.