How much patience is needed in a PhD?

The PhD is a test of your stubbornness, patience and grit more than it is a test of any sort of intelligence or ability.

Benjamin Franklin is said to have said "I haven't failed. I've found 10,000 things that don't work." It is all in the question of how you frame things. Abstract mathematics is not one of those publish-30-papers-a-year fields. It can take a horribly long time until you finally see how to solve your problem, and then you can't un-see it anymore and you wonder why you were being so silly about this for the past x-years. Just keep at it, find one more thing every day that doesn't work, and make sure to do something that is NOT mathematics every day.

As noted by Ian Sudbery above, the PhD just means you were able to convince a committee of busy professors that the complete sentences/experiments/proofs you have bound between two covers is an original contribution to your field.

Also, http://phdcomics.com/ helps. Daily. Read all the back strips, too.

As previously discussed we cannot answer your question, as we cannot quantify patience. But I really wanted to add another measure for you. Not saying you don't need patience (you need it, a lot, generally in life).

But what about resilience?

I would strongly advise you to think about fostering your resilience. It feels less hopeless and helpless than patience. But in the end, patience is probably a part of it. Further discussions of this would most likely be more in the scope of Philosophy.SE.