What qualities made Slughorn wanted by Voldemort?

Question

In Harry Potter and the Half-Blood Prince Dumbledore finds Professor Slughorn, who is in hiding. Slughorn tells Dumbledore that he’s been in hiding because the Death Eaters have been trying to recruit him for almost a year. I have two questions:

1. Why did the Death Eaters and Voldemort want to recruit Professor Slughorn?
2. If they didn’t want to recruit Slughorn, and wanted to kill him instead, was it because Slughorn knew about the Horcruxes and that Dumbledore was hunting the Horcruxes in order to destroy them?

Answer 1

First of all, it wasn't #2.

There are 4 proofs of that:

1. As discussed here in passing, if Voldemort wanted to kill Slughorn to keep hidden their Horcrux conversation - he could have done it anytime since his graduation to the start of First Wizarding War to the day he tried to kill Harry. Instead, Slughorn is ONLY in hiding at the start of Year Five:

   'I haven't given them the chance. I've been on the move for a year.' (from HBP, which is before start of Year Six).

   Remember that at the start of Year Five, Voldemort and DEs had no clue Dumbledore was aware of Horcruxes or hunting them.

2. Also, if Slughorn, the ultimate taker-care-of-self, was under threat of life, he'd be likely hiding out abroad, NOT in England.

   This is further proven by him stating that he worries about his safety if he joins Hogwarts faculty:

   'Still... the prudent wizard keeps his head down in such times. All very well for Dumbledore to talk, but taking up a post at Hogwarts just now would be tantamount to declaring my public allegiance to the Order of the Phoenix! And while I'm sure they're very admirable and brave and all the rest of it, I don't personally fancy the mortality rate –'

   He wouldn't have said that he if was running literally for his life.

3. Most importantly, both Slughorn AND Dumbledore seem to strongly imply the motive was recruitment:

   'So, all these precautions against intruders, Horace... are they for the Death Eaters' benefit, or mine?' asked Dumbledore.

   'What would the Death Eaters want with a poor broken-down old buffer like me?' demanded Slughorn.

   'I imagine that they would want you to turn your considerable talents to coercion, torture, and murder,' said Dumbledore. 'Are you really telling me that they haven't come recruiting yet?'

   Slughorn eyed Dumbledore balefully for a moment, then muttered, 'I haven't given them the chance.

4. And, for a direct proof, Slughorn admits this to Harry:

   ...'And I suppose one could argue that as I have not joined the Death Eaters, He-Who-Must-Not-Be-Named can hardly count me a friend... in which case, I might well be safer a little closer to Albus... I cannot pretend that Amelia Bones's death did not shake me... If she, with all her Ministry contacts and protection...'

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That leaves wanting to recruit him. Why?

There's no canon direct answer I'm aware of, BUT:

• Slughorn was an eminent potions master. He was Snape's Potions teacher. He could brew Liquid Luck (Felix Felicis).

  Dumbledore himself considered him posessing of "considerable talents". Dumbledore's guess (which as we know are usually right) was:

  'I imagine that they would want you to turn your considerable talents to coercion, torture, and murder

• Slughorn was extremely well connected, socially and politically. Having access to his LinkedIn social network would have been a great boon to Death eaters - money, influence, Imperioused influential figures.

Answer 2

Becuase of his past relationship to Voldemort and his skill with Potions

While Harry’s story was widely dismissed by both the Ministry of Magic and the wizarding press, Horace Slughorn believed it. Confirmation came three nights after the death of Cedric Diggory, when the Death Eater Corban Yaxley arrived at Slughorn’s house under cover of night, clearly intending to recruit him, or take him by force to Voldemort.

Slughorn reacted with a speed that would have astounded those who had watched him grow slower and fatter through the years of his retirement. Transfiguring himself into an armchair, he successfully evaded Yaxley’s detection. Once the Death Eater had left, Slughorn packed a few necessities into a bag, locked up his house behind him, and went on the run.

For over a year, Slughorn moved from house to house, often squatting in Muggle dwellings when the owners were away, because he did not dare stay with friends who might subsequently betray – whether willingly or under duress – his whereabouts. It was a miserable existence, made still more wretched by the fact that he did not know precisely what Voldemort wanted from him. He thought it most likely that his old student simply wanted to recruit him to his army, which was still small compared to what it had been at the height of his previous power; in his darkest moments, however, Slughorn wondered whether Voldemort did want to kill him, to prevent him ever betraying the source of the latter’s continuing invulnerability.
_{Pottermore - Horace Slughorn (behind paywall)}

While Slughorn briefly thought Voldemort wanted to kill him, this was later revealed to be false.

Once the school had been taken over by Lord Voldemort, with Severus Snape as Headmaster and the Death Eater Carrows taking key roles in subjugating staff and pupils, Slughorn learned that Voldemort had nothing worse in store for him than to remain in post and teach pure- and half-bloods. This he did, keeping his profile as low as he dared, though never enforcing the violent discipline advocated by the Carrows, and attempting to look after the students in his care as best he could.
_{Pottermore - Horace Slughorn (behind paywall)}

Answer 3

I agree that it seems unlikely that Voldemort would keep Slughorn alive given that he was the only one who knew of his plan. But I think Voldemort was a good judge of character and knew Slughorn would be too ashamed to admit his role to anyone. I think if Voldemort had found Slughorn when he was on the run he would have tried to recruit him and failing that he would've killed him.