Horace Slughorn's club, the Slugclub, was his club for "special students", but it was moreso his way of becoming chummy with kids that had high profile friends or family. So why was a muggle-raised, friend-less orphan like Tom Riddle allowed in?
Slughorn loves having connections, so sure, alot of the members of the slug club were relatives of famous witches and wizards, but what about the next generation? He also collected students who had the potential to do great things.
‘Horace,’ said Dumbledore, relieving Harry of the responsibility to say any of this, ‘likes his comfort. He also likes the company of the famous, the successful and the powerful. He enjoys the feeling that he influences these people. He has never wanted to occupy the throne himself; he prefers the back seat – more room to spread out, you see. He used to handpick favourites at Hogwarts, sometimes for their ambition or their brains, sometimes for their charm or their talent, and he had an uncanny knack for choosing those who would go on to become outstanding in their various fields. Horace formed a kind of club of his favourites with himself at the centre, making introductions, forging useful contacts between members, and always reaping some kind of benefit in return, whether a free box of his favourite crystallised pineapple or the chance to recommend the next junior member of the Goblin Liaison Office.’ -HBP
Tom Riddle is the most gifted student Hogwarts has had since Dumbledore, his potential was limitless. Slughorn even expected Tom to become Minister of Magic one day.
‘– I confidently expect you to rise to Minister for Magic within twenty years. Fifteen, if you keep sending me pineapple. I have excellent contacts at the Ministry.’-HBP
Lily Potter and Hermione are two other examples, exceptional witches in their own right.
Yes, well. You shouldn’t have favourites as a teacher, of course, but she was one of mine. Your mother,’ Slughorn added, in answer to Harry’s questioning look. ‘Lily Evans. One of the brightest I ever taught. -HBP
Another muggle born with potential Dirk Cresswell was also collected.
And there was Dirk Cresswell in the year after her, too – now Head of the Goblin Liaison Office, of course – another Muggle-born, a very gifted student, and still gives me excellent inside information on the goings-on at Gringotts!’ -HBP
Many of his connections include famous Quidditch players, which means he just snagged the hottest players on the house teams into his club for the potential pay off if they go pro.
He also helped set bright students up for success
And Ambrosius Flume, of Honeydukes – a hamper every birthday, and all because I was able to give him an introduction to Ciceron Harkiss, who gave him his first job! -HBP
Because Tom Riddle was a "special student" in terms of academic excellence.
'He reached the seventh year of his schooling with, as you might have expected. top grades in every examination he had taken. All around him, his classmates were deciding which job they were to pursue once they had left Hogwarts. Nearly everybody expected spectacular things from Tom Riddle, prefect, Head Boy, winner of the Special Award for Services to the School.
I know that several teachers, Professor Slughorn amongst them, suggested that he join the Ministry of Magic, offered to set up appointments, put him in touch with useful contacts. He refused all offers. The next thing the staff knew, Voldemort was working at Borgin and Burkes.'
I'd expect Tom, just like Hermione, did excellently in the Potions subject, and this definitely would have sparked Slughorn's interest.