Ron likely vaguely remembered Draco was connected with one.
There isn't any information on when Draco actually bought the Hand of Glory in the books, or on a logical reason why Ron would think he had it, although he did actually have it. This is pure speculation, but I've tried to figure out a logical in-universe reason for why he would think that.
Now, Ron wasn't there when Harry was in the cabinet and listening to the Malfoys' conversation with Borgin. He would have only possibly heard about it secondhand, if Harry told him what had happened. Harry did start telling Ron, Hermione, and the other Weasleys about being in Borgin and Burkes and seeing the Malfoys.
“Guess who I saw in Borgin and Burkes?’ Harry asked Ron and Hermione as they climbed the Gringotts steps. ‘Malfoy and his father.’
‘Did Lucius Malfoy buy anything?’ said Mr Weasley sharply behind them.
‘No, he was selling.’
‘So he’s worried,’ said Mr Weasley with grim satisfaction. ‘Oh, I’d love to get Lucius Malfoy for something …”
- Harry Potter and the Chamber of Secrets, Chapter 4 (At Flourish and Blotts)
Ron may have presumed that since Draco wanted the Hand of Glory, he got it. Although from what Harry saw, Lucius clearly said no to him, Draco usually does get what he wants eventually, especially if that thing is something purchasable by money as opposed to something that's not (like the courage to be a Death Eater). Draco expresses a clear desire to have the Hand of Glory when he's with his father at Borgin and Burkes. It's possible that Harry didn't explicitly tell Ron that Draco didn't buy the Hand of Glory, because Draco being interested in it was only a minor detail. At that time, it's unlikely that either of them would have thought that important enough to carefully consider. So it's possible Ron could have thought since then that Draco had the Hand of Glory.
“Can I have that?’ interrupted Draco, pointing at the withered hand on its cushion.
‘Ah, the Hand of Glory!’ said Mr Borgin, abandoning Mr Malfoy’s list and scurrying over to Draco. ‘Insert a candle and it gives light only to the holder! Best friend of thieves and plunderers! Your son has fine taste, sir.’
‘I hope my son will amount to more than a thief or a plunderer, Borgin,’ said Mr Malfoy coldly and Mr Borgin said quickly, ‘No offence, sir, no offence meant –’
‘Though if his school marks don’t pick up,’ said Mr Malfoy, more coldly still, ‘that may indeed be all he is fit for.”
- Harry Potter and the Chamber of Secrets, Chapter 4 (At Flourish and Blotts)
In Half-Blood Prince, when Ron says Draco might have broken his Hand of Glory, years have passed. It's possible that by that time, Ron would have forgotten the details, and just remembered that Draco and a Hand of Glory were connected at some point. At the time the Hand of Glory came up, Harry was trying to speculate on what Draco was up to in Knockturn Alley, and Ron and Hermione were trying to help him think of ideas. Ron probably thought of it because it was something he remembered connected with both Draco and Knockturn Alley.
“Harry spent a lot of the last week of the holidays pondering the meaning of Malfoy’s behaviour in Knockturn Alley. What disturbed him most was the satisfied look on Malfoy’s face as he had left the shop. Nothing that made Malfoy look that happy could be good news. To his slight annoyance, however, neither Ron nor Hermione seemed quite as curious about Malfoy’s activities as he was; or at least, they seemed to get bored of discussing it after a few days.
‘Yes, I’ve already agreed it was fishy, Harry,’ said Hermione a little impatiently. She was sitting on the window-sill in Fred and George’s room with her feet up on one of the cardboard boxes and had only grudgingly looked up from her new copy of Advanced Rune Translation. ‘But haven’t we agreed there could be a lot of explanations?’
‘Maybe he’s broken his Hand of Glory,’ said Ron vaguely, as he attempted to straighten his broomstick’s bent tail twigs. ‘Remember that shrivelled-up arm Malfoy had?”
- Harry Potter and the Half-Blood Prince, Chapter 7 (The Slug Club)