Answer:
Mikoto did it in hopes that her son would be a great shinobi just like his namesake.
She told this to Kushina and
Biwako Sarutobi, Hiruzen's wife.
The Uchiha had no beef with Hiruzen's family. Don't forget that the alienation and segregation towards the Uchiha clan was started by Tobirama and then further reinforced by
Danzo. These two were firm believers that separating the Uchiha clan from the rest of the village was for the good of the village though Danzo did it to greater extremes to the point where yes, there was a coup de'tat planned by Fugaku and particular senior members of the clan.
Don't also forget that Hiruzen was completely sympathetic to the Uchiha clan but Danzo pushed for more alienation of Uchiha rights and pretty much bullied the rest of the council to go about
his plans. And he didn't even have Shisui's eye yet at this time unless we assume that some time shortly after Shisui died was when he had it implanted. In any case, Danzo's grip on the council is why Itachi was given his orders by Hiruzen in complete secrecy.