Well.. i currenly believe in two theories regarding the gift Itachi gave to Naruto...
Theory 1 : This seems to be the most widely accepted theory and which i support too... that when the time naruto battles sasuke and itachi shows up(somehow, either from a genjutsu or some sort of seal like minato did) and convince sasuke.. to go against madara.. and the tell sasuke the most accurate version of the uchiha massacre..
But while reading another thread "my thoery on why naruto might unlock the sharingan".....
A person came up with an interesting thoery saying that 'Maybe Itachi gift was the same as Madara to Nagato? A way to awaken the Rinnegan?, at a critical point in a battle?'
What i thought was...
This could end up being a good theory.. but there is a plot hole.. well you see madara claims he gave nagato the rinnegan.. madara has the EMS.. itachi too has it.. and both madara and itachi are geniuses, so it could work itachi doing the same thing to naruto..(another supporting point is that nagato and naruto both were uzumaki.. and to add to that naruto has the body of the so6p).. but we dont know were sure if madara really did give or was he lying.. in the manga it is clear that nagato was distantly related to the sage in some way and the rinnegan has reawakened through him..
if whatever madara did really give.. how did he.. and why didnt he keep it to himself.. did he give to only take it back again??
But this theory seems possible too....