So far, canon only explains the diadem’s powers in detail. I’m not aware of a detailed description of the other artefacts, but we can make educated guesses.
We learn about Ravenclaw’s diadem when Harry et al. are trying to work out what Ravenclaw’s artefact might be, and Terry Boot provides your answer:
“Sorry, but what is a diadem?” asked Ron.
“It’s a kind of crown,” said Terry Boot. “Ravenclaw’s was supposed to have magical properties, enhance the wisdom of the wearer.”
— Deathly Hallows, chapter 29, The Lost Diadem
That fits with the general perception of Rowena Ravenclaw, and is consistent with the crown that Luna’s father has tried to recreate.
Since Voldemort has turned the diadem into a horcrux, nobody puts it on to test this theory, as it’s probably cursed (like the ring). Any magic from Ravenclaw is probably destroyed at the same time as the horcrux.
But the notion that it provides intelligence is confirmed when Harry talks to the Grey Lady, aka Helena Ravenclaw, who was around when the diadem was first created:
“While the diadem bestows wisdom,” she said with an obvious effort to pull herself together, “I doubt that it would greatly increase your chances of defeating [Lord Voldemort].”
— Deathly Hallows, chapter 31, The Battle of Hogwarts
This is when she explains that she stole the diadem to make herself wiser than her mother, and how the diadem fell into Voldemort’s hands.
All we have is the quote that you’ve already provided. We know it was powerful, but beyond that, nothing is confirmed.
Since Gryffindor’s sword and Ravenclaw’s diadem both embodied characteristics of their owner, perhaps the same might apply for Hufflepuff. But how does “goodness” apply to a cup? One answer might come from Hufflepuff’s background at Hogwarts:
One of the four celebrated Founders of Hogwarts, Hufflepuff was particularly famous for her dexterity at food-related Charms. Many recipes traditionally served at Hogwarts feasts originated with Hufflepuff.
— Wizard of the Month, June 2007, JKRowling.com
This caused quite a lot of discussion when it was first posted, and later J.K. Rowling had to clarify Hufflepuff’s role in the Hogwarts kitchens:
Hufflepuff did what was the most moral thing to do at that time, and we are talking about over a thousand years ago. So that would be to give them good conditions of work. There was no kind of activism there, so no one's going to say, “Here’s an idea, let’s free them. Let’s pay them.” It was just, “Well we'll bring them somewhere they can work and not be abused.”
— PotterCast interview J.K. Rowling, part one, December 2007
So food-related charms seem like a natural fit for her artefact, and of course that goes nicely with a cup. Perhaps it provides everlasting water, or healing properties. Again, these abilities are never explored, because Voldemort’s horcrux means that it probably provides poison or maggots instead.
I don’t think J.K. Rowling has ever commented on this, but the obvious comparison is the Holy Grail. That’s where I’m getting the idea that it provides some sort of magical drink.
Later, in the same memory that you quoted, we learn that Slytherin’s locket possess remarkable powers, when Hepzibah Smith shows the locket to a young Riddle:
“I had to pay an arm and a leg for it, but I couldn't let it pass, not a real treasure like that, had to have it for my collection. Burke bought it, apparently, from a ragged-looking woman who seemed to have stolen it, but had no idea of its true value—”
There was no mistaking it this time: Voldemort’s eyes flashed scarlet at the words, and Harry saw his knuckles whiten on the locket's chain.
“—I daresay Burke paid her a pittance but there you are…. Pretty, isn't it? And again, all kinds of powers attributed to it, though I just keep it nice and safe….”
— Half-Blood Prince, chapter 20, Lord Voldemort’s request
Given Slytherin’s background with snakes, it seems natural to assume that the locket might grant some sort of affinity with snakes. When Harry and Ron try to destroy the locket, Parseltongue is required to open it. That seems like a defence that might have been designed by Slytherin, and the locket’s powers could only be accessed if you could open the locket.
It’s not inconceivable that only Slytherin’s ancestors could open the locket (a la the Chamber of Secrets), but that’s pure speculation on my part.