I've done this several times, starting with the AD&D module N1 and continuing with many other AD&D and B/X & BECMI D&D modules. There are, broadly speaking, two types of conversion that need to happen; one is not too difficult, the other one can be VERY difficult.
First, the mechanical conversion, which is not too bad, usually. As others have posted, there's adjusting the monsters, traps, and treasure. Earlier monsters mostly have WAY fewer HP, so you usually have to adjust the number of monsters down a lot. They also tend to have very strong special abilities - sometimes deadly - and very weak normal attacks. Traps tend to be deadly, so you have to decide whether to keep them that way, or adjust them so they do enough damage that they MIGHT be deadly, rather than that they ARE deadly. Treasure is XP in (A)D&D, so you've got to drastically reduce the treasure when converting a module to 5e.
What's much more difficult IME is navigating the differences in underlying assumptions about (A)D&D vs. 5e. 5e is about adventure and combat. (A)D&D is about loot and survival. When you play AD&D, for example, you know that you need to carefully avoid or circumvent any traps because they will certainly kill your PC, whereas in 5e the expectation is that you will attempt to disarm a trap, but if you fail, it will do a bunch of damage. AD&D modules often involve riddles and puzzles, which rely on the players using their own metagame cleverness to solve. In 5e, where there is generally a stronger expectation that you "play only what your character knows," this sort of thing stumps PCs and frustrates players. The 5e death save system leads to a situation where one PC very rarely dies but TPKs happen much more often, compared to (A)D&D. How do you adjust a module for these things, given that they are underlying assumptions about the module?
A lot of the official conversions that others have mentioned in this thread only address the first part, but not the assumptions part. Playing through White Plume Mountain as I did with one of my 5e groups last year, I was struck again and again how tonally different that module is to any modern 5e play. Actually converting it so that it would make sense for 5e is ridiculously challenging. Do you try to make it as deadly as the original? Significant changes need to be made, much more than was done in TotYP. But should it be as deadly as the original, when 5e itself mostly isn't? How does a deathtrap dungeon hit when it isn't a deathtrap? It's very different.
