The other answers point out the mechanical burden well, so here are a few other considerations.
Regardless of when the personalities manifest, and what the player and the DM have to keep track of, there is also the party and flow of play to consider. Do all the personalities have the same alignment? Do they have access to the same memories? It could be jarring for the party to have a game plan going in to a long rest, only to find one of their members does not remember the plan, or worse just doesn't agree with it. This could add a lot of drag to group decision making.
Another big wrench in the whole thing, waking up with a different personality everyday is an unlikely manifestation of Dissociative Identity Disorder. The character would likely change personalities due to stress or other environmental triggers (which would make it even harder to play). Waking up with a different personality feels arbitrary, and probably is due to the fact that mechanically it's the only way to keep the access to spells and subclass features from being completely broken.
Assuming the best case scenario that the personalities have access to the same information, generally have the same values and opinions, and neatly wake up everyday with different abilities... why do they want to do it? Great characters are best built around a core personality where the abilities and stats are an expression of that personality. Trying to hack around the limitations of the rules, or avoid making a clear decision about what your character is (or is not) is going to lead to confusing and difficult roleplaying (if it's possible to roleplay at all). Maybe roleplay isn't a big concern for your player or the group, but in that case simple is better, and this character is not simple by any means!
In the end, you want your player to enjoy their character, but they might be happier with multi-classing if they are not ready to be decisive.