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Tom Riddle was an orphan. He never knew his parents. IIRC, the orphanage only knew the name his mother wanted to give him, not even her own name.
How did he trace his mother's family and find the Slytherin connection?
(I'm starting to vaguely remember something semi-related in the memory Dumbledore extracted from Morfin, but I can't remember the exact details)
Merope DID give some information to the orphanage. Enough, as it turns out.
From Half-Blood Prince:
“Did she say anything before she died?” asked Dumbledore. “Anything about the boy’s father, for instance?”
“Now, as it happens, she did,” said Mrs. Cole, who seemed to be rather enjoying herself now, with the gin in her hand and an eager audience for her story. “I remember she said to me, ‘I hope he looks like his papa,’ and I won’t lie, she was right to hope it, because she was no beauty — and then she told me he was to be named Tom, for his father, and Marvolo, for her father — yes, I know, funny name, isn’t it? We wondered whether she came from a circus — and she said the boy’s surname was to be Riddle. And she died soon after that without another word.
So, we have his Grandfather identified as Marvolo (we don't have 'Gaunt' yet), and his father as Tom Riddle.
Several chapters later:
“All he had to go upon was the single name ‘Marvolo,’ which he knew from those who ran the orphanage had been his mother’s father’s name. Finally, after painstaking research through old books of Wizarding families, he discovered the existence of Slytherin’s surviving line. In the summer of his sixteenth year, he left the orphanage to which he returned annually and set off to find his Gaunt relatives. And now, Harry, if you will stand …”
And, if he wasn't sure, once he got there, his resemblance to his father confirmed that he had come to the right place (this is the 'Memory extracted from Morfin' that you alluded to):
“I thought you was that Muggle,” whispered Morfin. “You look mighty like that Muggle.”
“What Muggle?” said Riddle sharply.
“That Muggle what my sister took a fancy to, that Muggle what lives in the big house over the way,” said Morfin, and he spat unexpectedly upon the floor between them. “You look right like him. Riddle. But he’s older now, in ’e? He’s older’n you, now I think on it. …”
. . .
“He come back, see,” he added stupidly.
Voldemort was gazing at Morfin as though appraising his possibilities. Now he moved a little closer and said, “Riddle came back?”
“Ar, he left her, and serve her right, marrying filth!” said Morfin, spitting on the floor again. “Robbed us, mind, before she ran off! Where’s the locket, eh, where’s Slytherin’s locket?”
