What was Itachi's mysterious illness? Do you think Tsunade could have cured it? I'm aware that it is a terminal illness. And when did Itachi get this illness?
The disease is never named.
This is actually quite common in manga/anime. It's often said that a character has some disease without the name of the disease ever being stated.
In some cases, the disease may be implied, such as in the case of said character suddenly being bald, in which case cancer is being implied. This can give a story some realism
In other cases, no symptoms are given or the symptoms are very generic, so that it's impossible to determine what disease the character has. This is done because it means less work on the author's part if they're not familiar with a disease that has the symptoms they want and means less time has to be spent developing things that don't have any relevance to the story.
In this case, very little information is given about Itachi's illness other than that it's terminal. It's likely that the author did this simply because what Itachi had didn't really matter to the story in any way. The only important thing is that it forced Itachi to speed up his plan, knowing he would die.
There is one other thing we know about his disease. According to Narutopedia, this disease had an effect on Itach's chakra. Because of this, it is reasonable to assume that this disease either isn't real or it's based on a real disease with this fictional characteristic attached.
Tobirama said in one episode that the Uchiha develop their Mangekyo Sharingan from knowing despair and great pain, and that their despair is reflected in how strong their eyes are.
Basically his illness is a rebound of his powers on his body; his body isnt able to handle the power without a new power to help control his eyes.
You can refer this link to learn more about Itachi
To be honest, i think that itachi may have never fully recovered from 8 trigrams attack he received from Kohinata Mukai as an anbu (itachi shinden,book 2,page 30) in that novel, itachi got hit with at least eight od the 64 palms technique. We dont know if hus internal organs or chakra flow ever fully recovered after that,and the tremendous strain from using mangekyo and other techniques was probably a bit too much with a deficient chakra flow. Every time he used chakra probably affected his vital organs somehow. This seems unlikely,simce i cant see him being unable to be treated, since hinata was able to recover after neji tried to kill her,but its the best theory i have.