From Harry Potter and the Sorcerer's Stone:
"Better Hufflepuff than Slytherin," said Hagrid darkly. "There's not a single witch or wizard who went bad who wasn't in Slytherin. You-Know-Who was one."
"Vol- sorry - You-Know-Who was at Hogwarts?"
"Years an' years ago," said Hagrid.
(Sorcerer's Stone, Chapter 5, Diagon Alley).
Hagrid knew that Voldemort went to Hogwarts. Did he know that Tom Riddle was Voldemort?
Maybe a casual acquaintance who met Tom while shopping at Borgin and Burkes would have a hard time identifying Tom Riddle after his transformation, but Hagrid, an (ex)schoolmate and later a Hogwarts gamekeeper was more likely to recognise Lord Voldemort as Tom, the Slytherin prefect who got an award for reporting Hagrid for the crime he didn't commit and causing his expulsion.
As the Keeper of the Keys of Hogwarts it is possible he even met Voldemort when Voldemort returned to Howarts at ~1966 and visited Dumbledore, reapplying for the DADA job. Maybe Hagrid even was the one let Voldemort in. From the description it seems that at that time Tom was still less snakelike and still recognisable:
Voldemort had entered the room. His features were not those Harry had seen emerge from the great stone cauldron almost two years ago: They were not as snake-like, the eyes were not yet scarlet, the face not yet masklike, and yet he was no longer handsome Tom Riddle. It was as though his features had been burned and blurred; they were waxy and oddly distorted, and the whites of the eyes now had a permanently bloody look, though the pupils were not yet the slits that Harry knew they would become. He was wearing a long black cloak, and his face was as pale as the snow glistening on his shoulders. The Dumbledore behind the desk showed no sign of surprise. Evidently this visit had been made by appointment. “Good evening, Tom,” said Dumbledore easily. “Won’t you sit down?” ~Harry Potter and the Half Blood Prince - Chapter 20 : Lord Voldemort's Request
Tom Riddle then had no restrains (re)introducing himself as Lord Voldemort, but if someone did identity him as Tom Riddle, he didn't deny it:
Voldemort did not answer at once, but merely sipped his wine. “They do not call me ‘Tom’ anymore,” he said. “These days, I am known as—” “I know what you are known as,” said Dumbledore, smiling, pleasantly. “But to me, I’m afraid, you will always be Tom Riddle.~ibid
So it is possible that Hagrid encountered Voldemort after he returned from his travels and was among the few who knew Voldemort was once called Tom Riddle.
I highly doubt that Dumbledore shared his suspicions with Hagrid about the identity of the Heir of Slytherin. After all, he had no proof. Not even when Hagrid showed up in Harry's second year, agitated, afraid that history may repeat itself and and the innocent (yet again) will be blamed :
...the door of the office flew open with an almighty bang and Hagrid burst in, a wild look in his eyes, his balaclava perched on top of his shaggy black head and the dead rooster still swinging from his hand. “It wasn’ Harry, Professor Dumbledore!” said Hagrid urgently. “I was talkin’ ter him seconds before that kid was found, he never had time, sir—” Dumbledore tried to say something, but Hagrid went ranting on, waving the rooster around in his agitation, sending feathers everywhere. “it can’t’ve bin him, I’ll swear it in front o’ the Ministry o’ Magic if I have to—” “Hagrid, I—” “—yeh’ve got the wrong boy, sir, I know Harry never—” “Hagrid!” said Dumbledore loudly. “I do not think that Harry attacked those people.” ~Harry Potter and the Chamber of Secrets - 12: The Polyjuice potion
Not even when the ministry officials came to take Hagrid to Azkaban did Dumbledore told them that back then Hagrid's accuser was in fact young Lord Voldemort aka Tom Riddle.
Dumbledore did not have proof of Voldemort's involvement until Harry returned from the Chamber with the Diary, and besides it seems he was pretty brisk with Hagrid, usually giving him simple instructions without lenghty explanations.
There are many possibilities beside the above mentioned chance encounter during Voldemort's visit: Hogwarts staff gossip, an overheard conversation (maybe even between Dumbledore and Slughorn), Voldemort's followers bragging about Slytherin being the best house the Dark Lord was in. .. or just some foggy rumors about You-Know-Who hailing from Slytherin House, without any specific information, which Hagrid shared with Harry when introducing him the wizarding world.
If Hagrid didn’t figure out the connection himself between Tom Riddle and Lord Voldemort himself, Dumbledore probably wouldn’t have told him because he wanted to keep his extensive knowledge of the Dark Lord’s past secret, so the Dark Lord wouldn’t suspect he knew of the Horcruxes.
“Sir, am I allowed to tell Ron and Hermione everything you’ve told me?’
Dumbledore considered him for a moment, then said, ‘Yes, I think Mr Weasley and Miss Granger have proved themselves trustworthy. But, Harry, I am going to ask you to ask them not to repeat any of this to anybody else. It would not be a good idea if word got around how much I know, or suspect, about Lord Voldemort’s secrets.”
- Harry Potter and the Half-Blood Prince, Chapter 10 (The House of Gaunt)
Dumbledore wanted to keep the knowledge of the Dark Lord’s past as quiet as possible so the Dark Lord was kept confident that his Horcruxes were safe and he didn’t need to secure them more. Although the Dark Lord was the reason that Hagrid was expelled, Dumbledore wouldn’t likely risk his grand plan by telling him the Dark Lord’s former identity.
Hagrid knowing the Dark Lord’s house doesn’t in itself prove he knew he was once Tom Riddle. It’s possible that Hagrid could have known the Dark Lord’s house without knowing his former identity. In another answer on how Ron (who definitely didn’t know who Tom Riddle was because he only recognizes it from an award) knew the Dark Lord’s house, I theorized that the Dark Lord may have stated or implied his house himself. Hagrid may know it the same way as Ron did.
What’s harder to tell is if Hagrid did figure it out himself. He never says anything that would indicate he actually did know. It is possible that he did know, however, depending on how much he was able to piece together from things he’d known about Tom Riddle and the Dark Lord.
I don't think so. In Half-Blood Prince when Dumbledore shows Harry the memories related to Voldemort he never mentions that Hagrid had any idea that Tom Riddle was Voldemort, even when they discussed the part about Riddle opening Chamber of Secrets and framing Hagrid for it. In fact hardly anyone ever knew about Tom Riddle being Voldemort.
"Very few people know that Lord Voldemort was once called Tom Riddle. I taught him myself, fifty years ago, at Hogwarts. He disappeared after leaving the school ... traveled far and wide ... sank so deeply into the Dark Arts, consorted with the very worst of our kind, underwent so many dangerous, magical transformations, that when he resurfaced as Lord Voldemort, he was barely recognizable. Hardly anyone connected Lord Voldemort with the clever, handsome boy who was once Head Boy here."
Chamber of Secrets; Ch 18, Dobby's Reward
Sorry I know I'm late on this haha. It's never clarified whether or not Hagrid was ever explicitly told, but I agree that Dumbledore likely wouldn't have told him as it changes nothing (Hagrid obviously already knows he was framed), and as people have mentioned, Dumbledore was keen on keeping the extent of his knowledge on the DL.
However, I do think it's fair to say Hagrid would have made the connection following Harry's second year. If not then, as someone mentioned before, he certainly would've known following Harry's speech at the end of the 7th book where he repeatedly calls Voldemort "Tom Riddle".
I saw people above saying that Hagrid knows nothing of the frame job, but remember when he's taken to Azkaban, he does tell them to follow the spiders. This implies that he knows the beast that the spiders refuse to speak the name of. True, Aragog says that they never told even Hagrid what the monster is. But following the explanation about what happened in the Chamber of Secrets and the revelation that Voldemort himself was Slytherin's heir controlling the monster it would be clear. Hagrid would then know that Voldemort/ Tom Riddle had opened the chamber 60 years ago, and had returned to reopen it.
I guess the one flaw here is the possibility that Hagrid wouldn't know the story, but it would surprise me if he never heard it at the time, or while in the Order of the Phoenix in the later years. I guess it's possible that this was one of those things Dumbledore would've wanted to keep quiet, but Harry freely admits to killing the basilisk in the 5th book when meeting about the DA. Also, Hagrid was aware of Harry facing off with Voldemort at the end of the first year, and was aware of each of the times following, so it leads me to think this story also would've floated around as part of Harry's lore within his circle.
If he does know, he certainly wasn't too worried about it. Seemingly would add a weird layer to Hagrid's fear of Voldemort, but I would think it would humanize him a bit.
