No, you can't learn a different cantrip, by PHB RAW.
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.