Prestige Classes get a bad rap... I always thought they were perfectly fine in 3.5 (the proliferation of feats, and particularly unintended feat combinations, were what ultimately drowned that system in bloat. It's telling that nothing really surpassed CoDzilla, but suddenly add Divine Metamagic...). At their best, they made interesting choices to help improve multi-class characters, or gave interesting mechanical bonuses to joining a specific organization.
That said... you don't need them in 5e.
Let's riff off of the comment from earlier about the three things prestige classes did:
I'm going to go with no.
Realistically, PrCs did only three things:
- Get most of the powers of a Base Class with an additional hyperspecialized powerset.
- Combine Two Base Classes progression, sometimes with a theme, because the normal multiclass rules tended to suck.
- Give a mechanical framework for some RP focused abilities that was of questionable practical use, and tended to make a character that is worse than if they had just stuck to the base class and actually roleplayed out that kind of stuff.
In 5e, Feats basically cover both of the first two bullet points. They're for multi-class dabbling (e.g, Arcane Initiate) or hyperspecialized abilitiies (e.g, Shield Mastery) and they do a pretty good job of that. There are also, of course the "multi-class" subclasses, like Eldritch Knight or Arcane Trickster, to say nothing of War Cleric (or really, any Paladin), blade-pact Warlock, etc. Both of these options are well covered.
That leaves us with the third bullet, which I will give the benefit of the doubt to as mostly referring to "organization" prestige classes. 5e has that covered too, from a variety of angles. Is the main benefit from a non-combat pillar? That's what background features are for. Here's a dirty secret; PCs can earn new background-like features as they travel. This is kind of what the faction system is designed to represent, but you don't need anything so convoluted. The characters gain new contacts, and call on specific favors at the appropriate time and locations.
What if the "organization" benefit is mechanical in nature? That is what Blessings and Charms are for. These are two different things, mechanically, but they don't really have to be. Basically, Blessings give a permanent passive benefit, while Charms give a temporary power with a set number of charges (or length of time) before expiring. But you could use this concept to gift players joining a specific organization access to a new permanent power.
Prestige classes would be too convoluted to fit into the current structure of 5e as-is, and with all that already exists I don't really see how there's a new for them.