How did Snape know that Moony was Lupin's nickname?

Question

So, Snape calls for Lupin because his name shows on the Marauder's Map.

“Mr. Moony presents his compliments to Professor Snape, and begs him to keep his abnormally large nose out of other people’s business.”

Snape froze. Harry stared, dumbstruck, at the message. But the map didn’t stop there. More writing was appearing beneath the first.

...

“So . . . ,” said Snape softly. “We’ll see about this. . . .”

He strode across to his fire, seized a fistful of glittering powder from a jar on the fireplace, and threw it into the flames.

“Lupin!” Snape called into the fire. “I want a word!”

Utterly bewildered, Harry stared at the fire. A large shape had appeared in it, revolving very fast. Seconds later, Professor Lupin was clambering out of the fireplace, brushing ash off his shabby robes.

“You called, Severus?” said Lupin mildly.

_{From PoA}

Obviously, Snape knows Lupin by nickname.

How did he find out? I imagine that the nicknames were more secret...

Answer 1

They used their nicknames in public.

A gang of chattering girls separated Snape from James, Sirius and Lupin, and by planting himself in their midst, Harry managed to keep Snape in sight while straining his ears to catch the voices of James and his friends. ‘Did you like question ten, Moony?’ asked Sirius as they emerged into the Entrance Hall.

—Harry Potter and the Order of the Phoenix

Slightly after:

Behind him, the Impediment Jinx was wearing off. Snape was beginning to inch towards his fallen wand, spitting out soapsuds as he crawled.

‘I wouldn’t go out with you if it was a choice between you and the giant squid,’ said Lily.

‘Bad luck, Prongs,’ said Sirius briskly, and turned back to Snape. ‘OI!’

—Harry Potter and the Order of the Phoenix

Snape was clearly in hearing range to know that James' nickname was Prongs. Whether he had ever heard the others nicknames doesn't matter to much since Prongs would have instantly linked the map to James, Lupin, Sirius, and Peter.

But Sirius seems quite cavalier about using their nicknames in public so its pretty safe to say he knew all 4 of their nicknames.

@everyone whose confused about Snape

‘Indeed?’ said Snape. His jaw had gone rigid with anger. ‘You think a joke-shop could supply him with such a thing? You don’t think it more likely that he got it directly from the manufacturers?’ Harry didn’t understand what Snape was talking about. Nor, apparently, did Lupin.

‘You mean, from Mr Wormtail or one of these people?’ he said.

‘Harry, do you know any of these men?’

[...]

‘Why did Snape think I’d got it from the manufacturers?’

‘Because ...’ Lupin hesitated, ‘because these mapmakers would have wanted to lure you out of school. They’d think it extremely entertaining.’

‘Do you know them?’ said Harry, impressed.

‘We’ve met,’ he said shortly.He was looking at Harry more seriously than ever before.

—Harry Potter and the Prisoner of Azkaban

Here, on top of the nicknames, the text in book 3 itself indicates that Snape believed Lupin to be the manufacturer to begin with, but just can't prove it.

Answer 2

First of all, we know that they used their nicknames in public while at school:

"Did you like question ten, Moony?" asked Sirius as they emerged into the Entrance Hall.

and later:

‘Bad luck, Prongs,’ said Sirius briskly.

so it's possible that Snape heard them and knows about the nicknames that way. Even if he didn't, though, Snape gets a rude message from each Marauder, so he knows there are four of them. We also know that they bullied him in school more or less as a group, and "some subset of the James-Sirius-Peter-Remus clique picking on Severus" is a fairly common occurrence:

Snape was on his feet again, and was stowing the OWL paper in his bag. As he left the shadows of the bushes and set off across the grass, Sirius and James stood up.
Lupin and Wormtail remained sitting: Lupin was still staring down at his book, though his eyes were not moving and a faint frown line had appeared between his eyebrows; Wormtail was looking from Sirius and James to Snape with a look of avid anticipation on his face.
‘All right, Snivellus?’ said James loudly.
[...]
Students all around had turned to watch. Some of them had got to their feet and were edging nearer. Some looked apprehensive, others entertained.

Also, they're not super original in their abuse:

‘How’d the exam go, Snivelly?’ said James.
‘I was watching him, his nose was touching the parchment,’ said Sirius viciously. ‘There’ll be great grease marks all over it, they won’t be able to read a word.’

and we know that this bullying is still a very sensitive topic for him. So when he gets a message like "Mr. Moony presents his compliments to Professor Snape, and begs him to keep his abnormally large nose out of other people’s business," he naturally thinks back to that time and those people. From the fact that the map was obviously made by (a) a group of four people who are (b) trouble-makers and (c) good at magic, and (d) like to make fun of his nose, it's a perfectly reasonable guess that those four are the James-Sirius-Peter-Remus clique, even if he doesn't know their nicknames.

From there, we have two options:

1. He guessed that "Moony" is Lupin, because he knows Lupin is a werewolf, affected by the cycles of the Moon, or

2. He just picked Lupin because while all the other member of the group were dead or in hiding, Lupin was conveniently next door.

Answer 3

Completely independent of the Harry Potter series, 'Lupus' is the Latin name for 'Wolf', and Moony is obviously referring to the Moon.

Also, as we see later that Snape knows that Lupin is

a werewolf,

so he may as well suspect that Lupin is the one.

Moon Wolf, QED.