Patience
Sirius doesn't have patience for much, but he can find some for becoming an animagus.
Sirius has little patience for anything. Little time to spend on one task, because there's so much he wants to do. He leaves projects unfinished (and allows Remus to wrangle him and make him complete them). He sometimes doesn't have the patience to even finish a chapter of a book at times because he's always thinking about what to do next. There's so much going on and there's so little time.
Even James has learnt that if a prank takes too long or is overly elaborate, Sirius is not the person for the job. Thankfully Peter and Remus are happy to do the studying and working out what to do and the time-frame. James sets the plan in motion, and Sirius does the things that don't take too long. The spell-work. He also takes the fall, more often than not.
But to carry a leaf in his mouth for an entire month? That's insanity. That requires a level of commitment and focus that Sirius knows he doesn't have. He expects to last a day. He wants to become an Animagus, but he wants an easy solution. But he came up with the idea and he's gotta give it a shot at least.
He puts the leaf in his mouth the day he arrives at James' house, two days into his moving there. He has little to say during those holidays, so it's as good a time to start as any.
He lasts a day. Two days. He reads the letter Remus sends after the full-moon and he writes a long letter back. He tells Remus in the letter of moving in with the Potters. That they'll talk about that when back at school. He asks about the full moon.
He doesn't mention the leaf.
Three days. Four. He hates this leaf. This is boring. He should try this another time. Maybe there will be advances in magic that make the process easier? Five. He's on the verge of taking the leaf out.
Then he gets the letter from Remus. The full moon was hard, it tells him. Broken ribs. Remus apologising for how long his reply took, because of course Remus would apologise for not writing a letter when he's got broken bones. Remus should never apologise for something like that. The letter didn't even take that long to arrive, but Remus worries too much. Remus - his family, because family is more than blood. Remus who fears that the others will discard him one day.
Remus who would do anything for any of them. Sirius would do anything for Remus. He knows he's never said that to his friend before, but maybe he can show Remus that he would.
The leaf stays. If Remus can do this every single full-moon, month after month, year after year, Sirius can keep a leaf in his mouth for a month.
It's for Remus. For Remus, he can do this.
The days pass and he writes letters. Another thing he focuses on. Maybe it's because he's so used to seeing Remus, but the Potters think Sirius needs the holidays to settle - that Remus and Peter can come over next time. He misses Remus. He thinks about Remus a lot. At first he tells himself that the leaf is a constant reminder. But a leaf wouldn't have him thinking about his friend's hair falling in his eyes or his friend's smile.
They reach the next full moon. They start the rest of the process. James' father finds the book in James' room but says nothing at all about it. But when the forecast predicts lightening, the Potters take a last-minute weekend away.
With thoughts of Remus, Sirius readies himself. Maybe he doesn't have the focus for long things, but he can find that focus for Remus.
And when the image of the dog forms in his mind - when he visualises his other form, seconds before his body shifts, he smiles to himself. Remus will never spend another full moon alone again - not if Sirius can help it, because he'll always be there to protect Remus, to make sure Remus never hurts himself.
And that knowledge - it makes this all worthwhile.