I DM'd the second session of our campaign this weekend and an amulet came up which is sort-of important in a future sidequest I've half hashed out, which may never actually come up. They found the amulet and first used detect magic on it, and then identify. The amulet was giving off weak enchantment magic, and then when they cast identify I had to come up with what it does on the fly.

I made it fairly vague - the amulet has something to do with controlling how others view you, and requires attunement from someone of a specific bloodline.

RAW, I feel like Identify should have identified exactly what the magic does - which I hadn't exactly decided yet - and that would have been detrimental to the game as I want to maintain some mystery about the thing!

How do you handle Identify when you have magical items which you want to remain mysterious?

Use Nystul Magical Aura. The Amulet has been subjected it to it to hide its true nature so Identify is effectively scrambled. To get the true abilities the party will need to cast Dispel Magic on it and then either use Identify again or Legend Lore dpending on how obfuscated you want it to be. The party likely won't think to Dispel Magic unless they recieve some clue so that should keep it mysterious and you can flavour the "attunement by a specific bloodline" as being a built in dispel magic effect so those who should be using it get teh full powers of the amulet.

How do you handle Identify when you have magical items which you want to remain mysterious?

I have actually done this in my campaign. I simply did not provide all of the details. I told them that Identify could reveal some of the object's abilities, but there may be even more that Identify seems unable to detect. The object is a pendant that one of the players inherited. It can do several things, but right now, all Identify told them was that it can do "Clairvoyance" once a day. He also uses it as his arcane focus (the character is a sorcerer).

At some point, he will find out what else it does (probably soon, based on where they are in the campaign) but it's been like 6 levels since they had it identified, and it has not done any more yet. (It's waiting for a trigger.)

As a DM, you have every right to do this sort of thing. It's not "against the rules" -- the object is an exception to the rules. There are always exceptions for story/drama purposes.

I've never done that only selective information using Identify based on its text. "If it is a magic item or some other magic-imbued object, you learn its properties and how to use them, whether it requires attunement to use, and how many charges it has, if any."  Obviously, if the item is cursed I'll never reveal that information but I feel, in my games where magic items are rare to find the party receives full instruction on what the item is capable of.  

It seems to me, if Identify doesn't reveal curses, that this implies that Identify does not strictly reveal *all* properties of the item. 

This gives the DM justification for concealing plot-specific magical properties. And IMO, plot-specific stuff supersedes the rules anyway. The OP is basically saying "I want to avoid spoilers." Just about any measure is acceptable, IMO, if you are trying to *avoid actual spoilers.* What's the fun of having the adventure or the plot spoiled?

Quote from BioWizard >>

It seems to me, if Identify doesn't reveal curses, that this implies that Identify does not strictly reveal *all* properties of the item. 

This gives the DM justification for concealing plot-specific magical properties. And IMO, plot-specific stuff supersedes the rules anyway. The OP is basically saying "I want to avoid spoilers." Just about any measure is acceptable, IMO, if you are trying to *avoid actual spoilers.* What's the fun of having the adventure or the plot spoiled?

 I should have clarified in my post I was speaking more for the magic items found in DMG.  I agree with your statement as it is a fair point in terms of plot-specific magical items targeted by Identify.   

I'm just trying to get this straight in my head.  I think you all covered this.  The primary use to the Identify spell right now is to give you the same information as you get from a Short Rest in a smaller amount of time.  Instead of a full hour, it takes 10 minutes, or if you care to, your can cast it as a Ritual, and it will take 20 minutes.  The spell is rather niche, as the only thing that it does is tell what spells might be on a creature.

The Identify spell is effectively just a relic, something left over from previous editions of the game.  The same is true of the Legend Lore spell.  Myself, I remember those long ago days of yore when the game was First Edition Advanced Dungeons and Dragons.  Back then, players only got the same options that people without Identify, Legend Lore, get now.  They could use the item, say by putting it on, or speaking aloud a word carved on the side of it, or they needed someone to cast Identify or Legend Lore.  The joker in the deck was that Cursed items could trigger if you used an item, or tried to cast Identify or Legend Lore on it.

That could be pretty risky.  You could end up at ground zero for a Fireball from that Wand you tried to use, or be stuck with it until the Curse was removed.

So your next option was the Bard.  This is what Wikipedia has to say about them.

"Bards in First Edition AD&D were a special class unavailable for initial character creation. A character could become a bard only after meeting specific and difficult requirements, achieving levels in multiple character classes, becoming a bard only later. The process of becoming a bard in the First Edition was that a human or Half-Elf had to begin with very high ability scores: Strength 15+, Dexterity 15+, Constitution 10+, Intelligence 12+, Wisdom 15+, and Charisma 15+. These daunting requirements made bards one of the rarest character classes." (Insert by me here) "When you were using 3d6 to generate Scores."  "Bards began the game as fighters, and after achieving 5th level (but before reaching 8th level), they had to change their class to that of thief, and after reaching 5th level as a thief (but before reaching 9th level), they had to change again, leaving off thieving and begin clerical studies as druids; but at this time they are actually bards and under druidical tutelage.

Bards gained a limited number of druid spells, and could be any alignment that was neutral on at least one axis. Because of the nature of dual-classing in AD&D, bards had the combined abilities of both fighters and thieves, in addition to their newly acquired lore, druidic spells, all level dependent druidic abilities, additional languages known, a special ability to know legendary information about magic items they may encounter, and a percentage chance to automatically charm any creature that hears the bard's magical music. Because bards must have first acquired levels as fighter and thief, they are more powerful at first level than any other class."

This version of the bard is a druidic loremaster, more than a wandering minstrel and entertainer, though the bard does have song and poetic powers as well."  It took 10 levels at minimum to become a Bard.  That's roughly Tier 3.  Bards had an ability called "Legend Lore", which became the spell we know and love today.

The final option was to consult a Sage, that was a special "class" of Non-Player Character.  They could do everything that a Bard could do, and suffered from the same risks.  That meant they were very expensive, and they took a lot of time to get you an answer.  The current deity in charge of the Weave in the Forgotten Realms was once a humble Sage named "Elminster".  He's come a long way since then.  Sages could use something called "Legend Lore" too.

So then, specific to the topic.  For any standard item from the DMG, the Identify should tell you everything you get from a Short Rest.  The spell Legend Lore might be able to tell you move than this.  It doesn't say, but you would think that information about what an item did, how it worked, what it take to use, and whether or not it was Cursed would turn up as "secret lore that has never been widely known."

When dealing with a Mysterious Item created as Homebrew, Identify shouldn't tell you anything. The DM might let you use the spell Legend Lore, unless the item is an Artifact, and has that kind of rarity.  Those normally don't show up until Tier 4, but a DM can put them in a Tier 1 game if they feel like it.  Or of course, the players could have their characters try and consult with a Sage.  Note that Sages don't get the Legend Lore spell.

I am tempted to give my Tier 1 players a Ring of Invisibility, just to watch and see what they do.

Personally, I avoid introducing any magic items without knowing how they work, for specifically that reason.

That said... I generally think of Identify as revealing the in-game, mechanical uses of the object. Whether it grants a +1 to DEX or requires attunement, etc. If, for example, it's a key in regards a quest line at some point... like, let's just say that it allows a specific NPC to access their locked away powers if that specific NPC attunes to it, casting Identify would just reveal something like, "It seems to be a component of a larger enchantment, but has no unique powers of its own". 

Identify for me does not always tell you what a magic item will do, it can tell you if it is magical, and it might give the caster a sense of the school of magic the enchantment is linked to or the type of effect it might have, but it is not fool proof and it is dependant on the caster understanding what they are interacting with. It isn't a sentient spell, it is a spell through which the caster can see flows and patterns in the magic within an item and, if they have the experience and knowledge, they can decipher what that magic is an how it might come to life. 

In terms of in game narrative it might be something like this. 

"You sit down, concentrate, your pearl in front of you, focussing on the item as the spell takes effect it starts to glow with an inner light, patterns flow across the edge of the blade of the sword, as you study them you see that it is one continuous pattern, no break, no gap, magic interlaced along the edge of the blade seeping into the main body of the sword. You get a sense this sword will never dull, never lose it's edge and will cut through anything. " 

With that I hand over the +1 sword magic item to the wizard. 

"You study this strange decanter, as the magic of the identify spell seeps into it the glass turns blue, no the interior turns blue, it contains you see a portal to another plane swirling within it focussing harder you see this is the water plane, reaching out you touch it and words appear in your mind, words of command"

With that I hand over the decanter of Endless water. 

Now in your case 

"As you touch the amulet and cast the spell the amulet comes to life, lighting up, it is infused with magic, but as you focus you find the truth to that magic slipping from you, tiny intricate patterns form in the surface, tiny runes appear to your magical eye, but as you look and concentrate their purpose and reason are blank to you. You know this is magical but, right now, it is beyond your understanding." 

Later on have the wizard find a book detailing the enchantment they saw when using the spell, or something similar to it, armed with this new knowledge the wizard can maybe try the spell again, or, remembering what they saw when they first cast it, they can now decipher the meaning and explain how the enchantment works.