Supposedly everything Itachi did was for the best of Sasuke. To keep him alive, to give him a purpose to survive and get stronger, etc. But he also asked him to kill his best friend to get the Mangekyou Sharingan. If he wanted the best for Sasuke, why would he ask him to kill his best friend, when this doesnt seem to have any apparent purpose for the best of Sasuke, and instead the idea would traumatize him more?
Considering the end result was Sasuke rejecting Mangekyō, Itachi may have told Sasuke how to get the Mangekyō Sharingan in an attempt to bias Sasuke against it. Itachi has been shown to be a very perceptive and conniving person. Examples of this are abundant:
- He sets a trap inside of Naruto to use Shisui's Sharingan to reprogram Sasuke to defend the Leaf Village. He ultimately ended up using it to reprogram himself while reanimated.
- He set a trap inside of Sasuke's eye to incinerate Tobi should the two cross paths.
- His interaction with Sasuke after slaughtering the Uchiha Clan was done to manipulate Sasuke into eventually killing him, with the goal of making Sasuke a hero of the Leaf Village.
- In the fight with Kabuto, Itachi was quick to discern how Kabuto's nature as a spy was similar to his own. And then he set a trap by manipulating Kabuto into performing certain actions, so he could use the Izanami, which ensnared Kabuto in a time loop until he became the person Itachi wanted him to be.
- He threatened Danzo in order to ensure Sasuke's safety. He obviously understood Danzo well enough to know this would keep Sasuke safe. After the Third Hokage's death, he specifically made an appearance to remind Danzo of their arrangement.
Seeing all this, it isn't entirely far-fetched to suppose Itachi may have foreseen the outcome of telling Sasuke about the Mangekyō Sharingan. And the outcome was indeed that Sasuke rebelled, disgusted by the idea of gaining power the same way as his brother, ultimately choosing to leave Naruto alive.
But why might Itachi do this instead of, for instance, just leaving Sasuke in the dark about this secret? He may have seen it as an inevitability that Sasuke would find the Uchiha Clan's secret stone tablet, in which case he had to do something to make Sasuke not want to try and get the Mangekyō. Well, when someone you hate tells you to do something, it tends to make you not want to do it.
This is just some speculation, and I don't necessarily believe it myself, but I thought it was worth delving into as an alternative to the other answers.
As an alternative to what I said, I'd first like to point out that not everything Itachi did was motivated by his desire to help Sasuke. If that were the case, he never would have slaughtered the Uchiha clan, which was a seriously traumatic event that Sasuke was never going to get over (and indeed, this one event basically colored Sasuke's entire life experience).
In that case, it is prudent to note that Itachi was also motivated by a desire for stability in the Leaf Village. In order to do that, for there to be no repercussions against the Leaf Village when he carries out his mission, he had to be the bad guy. And in that case, he has to make his motives seem believable. Thus, he tells Sasuke about Mangekyō not for Sasuke's benefit, but as a clever cover, as if to say, "By the way, do you want to know why I killed Shisui?"
Itachi's stated goal was to make himself the villain, and Sasuke the hero for defeating him. Sasuke's not going to be a hero in the Leaf Village if he kills his best friend (who would presumably be a Leaf Shinobi). This further suggests my first theory might be correct (these two theories aren't mutually exclusive, as Itachi may have multiple motives for doing things).
Again, Itachi had to be the villain. Thus, Itachi and the Leaf higher ups left a lot of evidence that he was the bad guy, for the other Shinobi, and for Sasuke. Danzo, for instance, upon arriving at the scene of the massacre, immediately declared that Itachi secretly harbored a deep hatred for his clan. And they did a good job because the result was that no one ever suspected a thing. Even Anbu like Kakashi, who'd been helping spy on the Uchiha and knew about their discontent, and who personally knew Itachi, never thought that Itachi was under orders. A shocked Kakashi instead concluded that even after all that time working together, he'd never really known Itachi.
Additionally, we have to remember Itachi is playing a role. If we're thinking on the role of villain Itachi, with his motives, it makes sense for him to tell Sasuke about Mangekyō. This though it seems it isn't necessary for Eternal Mangekyō, it still seems like it would have been preferable. In which case, Itachi, acting out his role, wants Sasuke to get Mangekyō. And I'll note that having a secret motive adds plausibility for everything Itachi did, so it's good he didn't just tell Sasuke, "Hey, I want you to get Mangekyō so I can take your eyes and make myself more powerful," but instead, revealed it to him later. Because at that point, Sasuke would be like, "I'm older now, and I have questions about the Uchiha massacre." But hearing Itachi wants his eyes, he'd be like, "Oh, now I get it."
It's possible that Itachi wished for Sasuke to become stronger. Perhaps because of the danger Danzo posed, he wanted Sasuke to have the Mangekyō. He may have felt it was necessary for Sasuke to have Mangekyō to defend himself. Or more likely in my opinion, he simply wanted Sasuke to be informed of the possibility. Being a spy and confused about what's right, Itachi may have been uncertain whether or not Sasuke should get Mangekyō. He wanted Sasuke to decide for himself. In which case, he told Sasuke while thinking something like this, "Well, Sasuke, I don't know what you should do, but if you want power, here's how to get it."
As I mentioned in the comments:
Maybe so Sasuke could get the power he wanted to kill Itachi? Not sure. Will have to research. Itachi pretty much lived to die at Sasuke's hand. (emphasis mine)
Taken from the Naruto wiki article on the Mangekyo Sharingan:
A Mangekyō Sharingan is distinguished from a normal Sharingan by its appearance, which changes the form of the tomoe seal. While the exact design differs for each user, they all resemble pinwheels. It is initially awakened by the trauma suffered from witnessing the death of someone close to the user.
I also pointed out in the comments that this can also be awakened by killing a close friend. Many Uchiha during Madara's era purposely did this after hearing how this eye was awakened in order to see if they could get it themselves. This process was succesful, which proves the Mangekyo Sharingan can also be awakened by killing a close friend, aside from having to see them die.
This fact is quite popular already, but it's getting spoiler tagged because it is far-ish in the series. Sasuke figured out that Itachi killed the Uchiha alongside Obito on the order of Konoha's elders. This was not intended.
Itachi wanted to die by Sasuke's hand. He said himself he deserved to be dealt with by another Uchiha. Even if he wanted the best for Sasuke, he still wanted him to kill his friend in order to give him the power of the Mangekyo, so Sasuke would be strong enough to kill him, as he always wanted to. This would fulfill Sasuke's long goal of revenge, with hopes of him not finding out about the Uchiha incident. To answer the question, although Itachi always wanted the best for Sasuke, he wanted him to kill his best friend to:
- Inform him about the Mangekyo Sharingan and how to awaken it
- Give him the power needed in order to get revenge, with Itachi intending for Sasuke not to learn about the Uchiha incident
Yet another example of Itachi being selfish to "help" Sasuke :D
Itachi knew the only way Sasuke would obtain the Mangekyō Sharignan would be by killing his best friend which would make Sasuke about 10-20 times stronger than he already is. Itachi wanted Sasuke to be strong enough to face any challenge in the cruel shinobi world they live in and he knew having the Mangekyō would give Sasuke a huge advantage which it obviously did. Shortly after attaining Itachi's Mangekyō Sharingan, Sasuke was even strong enough to take on the Raikage at such a young age.
You cannot assume Itachi wanted Sasuke to kill his best friend. All we know is that he told Sasuke how to get the MS and that is by experience the intense emotion of seeing someone you love die. This is ultimately how he did obtain it after watching Itachi die.
Itachi knew that a strong Sasuke was important for the future of Konaha. He also knew that there was literally no one else who could teach this to him. Seeing as when Kakashi shows Itachi that he also has an MS (although an incomplete one) Itachi is shocked and astounded. Noting that this is to be expected from a Konaha shinobi.