Lord Voldemort is known to most of the wizarding world as "Lord Voldemort."
Would he have been named this on the Marauder's Map or would he have been called "Tom Riddle"?
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The map in the film appears to shows the person's chosen name, not their given name. That's why Newton Scamander's name shows up as Newt.
Presumably "Lord Voldemort" would show up with that name, rather than "Tom Riddle."
The map in the book appears to shows the person's given name, not their chosen name. In Harry Potter and the Goblet of Fire, Harry spies "Bartemius Crouch" in Moody's rooms. Had it been using Barty Crouch's preferred name, it presumably would have shown up as "Barty Crouch".
Presumably "Tom Riddle" would show up with that name, rather than "Lord Voldemort".
Peeves was not the only thing that was moving. A single dot was flitting around a room in the bottom left-hand corner – Snape’s office. But the dot wasn’t labelled ‘Severus Snape’ … it was Bartemius Crouch.
Harry Potter and the Goblet of Fire: CHAPTER TWENTY-FIVE — The Egg and the Eye
I'll say "Tom Riddle" for one very simple reason: Voldemort was in Hogwarts for an entire year, during which the twins had the Marauder's Map. It's understandable that Fred and George would see the name "Tom Riddle" right next to Quirrell's and not bat an eye (just like they did with "Peter Pettigrew"). But "Lord Voldemort" in the castle would HAVE to be noticed.
And as others have pointed out, people like Fake Mad-Eye and transmogrified Sirius and Peter show up under their real names on the map. Hard to see why Voldemort would come up as an alias.
I can't see that the Map would be able to distinguish between someone's birth name and any subsequent aliases that they might've adopted. It is just a map; it has no way of telling what someone likes to be called and no awareness of the huge cultural impact surrounding a name like 'Voldemort'. I don't think the Map cares much for nicknames or aliases; its magic simply writes the first name and surname of each individual, regardless of who that person is or what name others may use to refer to them.
The most telling example we have to demonstrate this is that of Sirius Black. Black also goes by the alias/nickname Padfoot. Yet on the map he is labelled with his full name.
“And then I saw another dot, moving fast toward you, labelled Sirius Black..."
(Prisoner of Azkaban, Chapter 17, Cat, Rat and Dog).
If the Marauder's Map was unwilling to ere from its First Name-Surname format for one of its own creators then I doubt that it would do so for anybody else. The natural conclusion is that 'Tom Riddle' would show up on the Map.
With Voldemort, there's more than just the name to consider. Harry never jumped in panic when he found his own name on the map, so we know the horcruxes didn't register. Maybe the map only shows complete souls; I'd guess that it can't resolve a soul fragment next to one that's intact.
As to whether the main bit of Voldemort had something the horcruxes didn't, or was "bright" enough to register on the map when nobody was around, I don't know, but I'm sure Voldemort would have made himself unplottable in any case.
He would most likely appear on the map as Lord Voldemort. as user Valorum mentioned, the map in the movie appears to use a person's chosen name, not their given name. Unlike Peter Pettigrew, who was actively trying to hide, nothing about Lord Voldemort's name is him trying to hide from who he is, it is representative of who he is.