Let's say I am a spellcaster with the Mage Hand cantrip. Then I multiclass into Rogue for 3 levels and pick up the Arcane Trickster subclass.
Cantrips: You learn three cantrips: Mage Hand and two other cantrips of your choice from the wizard spell list. At 10th level you learn another cantrip from the wizard spell list.
Is this Mage Hand redundant? Must I choose a different cantrip to begin with if I don't want redundant spells?
When you can choose a different "thing" in the case you already know or own the intended one, it should be specified, e.g. for skill proficiencies in background
If a character would gain the same proficiency from two different sources, he or she can choose a different proficiency of the same kind (skill or tool) instead.
Usually I would blame this on the fact that the PHB is written supposing you are not multiclassing, and the multiclassing section is short and quick, letting pass some details. Sadly, this might not be the case here, since you could get Mage Hand through the High-elf race feature Cantrip, so you don't need to be multiclassing for it to become an issue.
Therefore, if you could learn something else instead, it would be stated explicitly.
Talk to your DM beforehand. By your wording, you didn't even choose your cantrips for your first spellcaster class. Talk to the DM before you choose it, so you know if he will let you choose another cantrip when the time comes or if you will just lose a cantrip because rules say so.
This question now has a new answer!
On November 4th 2019, the Class Feature Variants Unearthed Arcana was released.
This UA includes a variant rule for every spellcasting class that has cantrips, that allows them to swap them out, either on long rest or on level up.
Bards, Sorcerers, and Warlocks can now swap them out on long rest:
Spell Versatility Whenever you finish a long rest, you can replace one spell you learned from this [Spellcasting/Pact Magic] feature with another spell from the [bard/sorcerer] spell list. The new spell must be the same level as the spell you replace.
Since cantrips are spells, this means you can swap cantrips out for other cantrips. This is specifically called out as possible in the same UA:
Rule Tip: Cantrips Are Spells Cantrips are 0-level spells, which don’t use spell slots. When a feature applies to spells, that feature applies to cantrips, unless the feature specifies that the spells must be of 1st level or higher or must expend a spell slot.
Clerics, Druids, and Wizards can swap them out on level up:
Cantrip Versatility Whenever you gain a level in this class, you can replace one cantrip you learned from this Spellcasting feature with another cantrip from the [cleric/druid/wizard] spell list.
So the spellcaster in question might only have to wait one long rest to get rid of their redundant cantrip. Or perhaps they will have to wait til they level up. But in either case, they won't be stuck with the redundant cantrip forever.
To be clear, this does not allow the Arcane Trickster to swap out their Arcane Trickster mage hand, but if they have mage hand from another class, that is what they can swap out instead. This is a better option anyway, since the Arcane Trickster's mage hand is superior to a normal mage hand.
As always, ask your DM to confirm that they are okay with you using these variant rules.
