Yes, but only 1st-level spells
PurpleVermont has already argued why such a multiclass could only write down spells of a level they could cast as a Wizard (so actually up to 2nd level, if they take Wizard 3). I disagree with Vermont's argument that stolen spells could not be transcribed at all, however.
The rules don't clearly state that you have to have seen a spell written down to be able to transcribe it, that's a matter of interpretation--and I disagree with the interpretation:
Copying a spell into your spellbook involves reproducing the basic
form of the spell, then deciphering the unique system of notation used
by the wizard who wrote it. You must practice the spell until you
understand the sounds or gestures required, then transcribe it into
your spellbook using your own notation.
The fact that different wizards use different notation implies that the written form of the spell is not an inherent part of the magic, merely an aide-memoire. And the requirement to practice the spell until you understand its V/S requirements -- precisely the thing which the Arcane Trickster can instantly intuit -- strongly suggests that this is the most important part of "learning" a spell, and the rest is merely paperwork that your first-level Wizard training would absolutely prepare you for.
The rules, as written, don't say.
There are no specific rules regarding the contents of an invisible mage hand but there is evidence that objects held in the hand may be invisible.
While we don't have many general rules regarding invisibility (mainly just what's in the "Hidden" sidebar in the PHB), we do have an invisibility spell and an invisibility monster action to base our decision off of. We also have usability to consider; why would the designers give us the ability to turn the mage hand invisible if it gives away its position as soon as it picks something (anything) up?
Invisibility the spell says that all objects worn or carried by the target are invisible. This seems fairly harmless and renders the spell kind of useless if it doesn't give completely invisibility to all objects carried and worn.
Under the control of a skilled arcane trickster, the mage hand can be used to manipulate objects, pick locks, pick pockets, or carry an object or objects weighing up to 10 lbs. It can also turn invisible, per the arcane trickster's class feature.
With that in mind, we can probably make some assumptions about the hand:
- It is probably the size of an average person's hand
- It is dexterous, or at least as dexterous as its controller
- It can manipulate small objects such as a lockpick
- It can steal things from people
We also know the following:
- In other cases in the rules, invisibility turns held objects invisible. This alone is not reason enough, but because it makes the spell invisibility useful (otherwise enemies would see your sword and attack you sight unseen) it is important to the argument (usability, as I mentioned above).
- The arcane trickster is a class archetype based around a rogue who uses magic to play tricks and get away with his roguish activities.
- The arcane trickster gets special rules for his mage hand.
Taking these things into consideration, it seems quite in line with the design goals of 5e and the overall flavor of the class to allow objects held in the hand and completely obscured by it to become invisible. This includes as many coins as can fit inside the closed fist, a dart, dice, a key, lockpicks, or other small, "palmable" objects. Larger objects, such as weapons, planks of wood, mugs of ale, and so on, would be visible because they are not completely obscured by the mage hand.
My reasoning for this:
- Invisibility the spell and invisibility the monster action generally grant invisibility to the target and all items worn or held.
- The arcane trickster relies on subterfuge and trickery. Why give them an invisible mage hand if it can't hide anything inside it? That's like saying, "Here's a beer, but don't drink it!" Or maybe more like, "Have this beer, you can drink it, but only when no one is looking. Oh yes and you're in a crowded marketplace."
- For the skeptical DM: It doesn't hurt anything to allow this! In fact, it will probably make your games more fun when your Arcane Trickster's player is having more fun.
- It isn't broken. It doesn't imbalance anything in the gameplay and it's easily overcome or made up for in other areas by a good DM. Personally, I don't think it even needs to be "made up" for. It is by no means a showstopper.
Although many tables allow some degree of tweaking in order to make a build match a concept, they usually involve some degree of horse trading, i.e. giving up something to get something else. Assassins get the Assassinate ability at third level. Arcane Tricksters get Mage Hand Legerdemain. Wizards get their own specialties. Unless he is willing to give up something from one of his other classes in order to justify the Legerdemain, you are certainly on firm ground to call him on this. Getting something for nothing is the definition of unbalanced.