There are two things that have to happen before a student has a tune thoroughly memorized. They have to have the melody of the tune so well in their head that they could hum it, and they need to be able to remember and physically execute the finger and bow patterns that the tune requires.
Some students will only have to hear a tune two or three times before they have it pretty well in their head, at least for short-term learning. Others may need to hear it a hundred times. This often depends on a student’s past musical experience, but not always. I‘ve taught students who have never sung or played an instrument who have developed great memories for tunes very quickly, as well as violinists who have never memorized before and take months to learn their first tune.
I like to tell my students that if they need to hear a piece 30 times to really get it memorized, they can do that by either playing it 30 times or by listening to it 30 times. Obviously, listening is a lot easier and they can combine it with other things like driving, eating, cooking, exercising, cleaning, or falling asleep. This doesn’t mean that they don’t have to practice, but they can cut down on time and frustration by doing this listening before or while they are learning a tune.
I usually suggest that a student should expect the first three tunes, however easy they are, to be difficult to learn and remember, and after that it will get easier. After learning 10 tunes, even easier and so on. I’ve probably learned 500 tunes at some point during my life and it keeps getting easier to learn a new tune (although harder to remember them all!).