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We know that the water of the Dead Sea is very much salty, so much so that no fish exist in it.

Is there any midrash or agada that speaks on the history of the Dead Sea, how it started to become salty?

According to the Midrash Tanchuma (Lech Lecha #8) the Dead Sea apparently became salty due to the sins of those involved in the war of the four kings against the five kings.

הוא ים המלח שבעונם נעשה אותו עמק למים מלוחים וכן הוא אומר ארץ פרי למלחה למה מרעת יושבי בה

It is the Dead sea, for in their sins that valley was made into salty waters. And so it says (Mechon Mamre translation) "A fruitful land into a salt waste, for the wickedness of them that dwell therein."

In Joshua 3:16 in the Bible, the Dead Sea is mentioned by the name “the sea of the Plain” and 'the Salt Sea', but no reason is given for the name 'the Salt Sea'.

14 When the people set out from their tents to cross over the Jordan, the priests bearing the ark of the covenant were in front of the people. 15 Now the Jordan overflows all its banks throughout the time of harvest. So when those who bore the ark had come to the Jordan, and the feet of the priests bearing the ark were dipped in the edge of the water, 16 the waters flowing from above stood still, rising up in a single heap far off at Adam, the city that is beside Zarethan, while those flowing toward the sea of the Arabah [the Plain], the Salt Sea [the Dead Sea, NRSV], were wholly cut off. Then the people crossed over opposite Jericho. 17 While all Israel were crossing over on dry ground, the priests who bore the ark of the covenant of the Lord stood on dry ground in the middle of the Jordan, until the entire nation finished crossing over the Jordan.

Josephus in Antiquities of the Jews Book V, Chapter 1.3 describes the same miraculous incident, but he makes no mention of the Salt Sea:

3. Now while Joshua, the commander, was in fear about their
   passing over Jordan, for the river ran with a strong current, and
   could not be passed over with bridges, for there never had been
   bridges laid over it hitherto; and while he suspected, that if he
   should attempt to make a bridge, that their enemies would not afford
   him thee to perfect it, and for ferry-boats they had none, - God
   promised so to dispose of the river, that they might pass over it, and
   that by taking away the main part of its waters. So Joshua, after two
   days, caused the army and the whole multitude to pass over in the
   manner following: - The priests went first of all, having the ark with
   them; then went the Levites bearing the tabernacle and the vessels
   which belonged to the sacrifices; after which the entire multitude
   followed, according to their tribes, having their children and their
   wives in the midst of them, as being afraid for them, lest they should
   be borne away by the stream. But as soon as the priests had entered
   the river first, it appeared fordable, the depth of the water being
   restrained and the sand appearing at the bottom, because the
   current was neither so strong nor so swift as to carry it away by its
   force; so they all passed over the river without fear, finding it to be in
   the very same state as God had foretold he would put it in; but the
   priests stood still in the midst of the river till the multitude should be
   passed over, and should get to the shore in safety; and when all
   were gone over, the priests came out also, and permitted the current
   to run freely as it used to do before. Accordingly the river, as soon
   as the Hebrews were come out of it, arose again presently, and carne
   to its own proper magnitude as before.

Bibliography: 'The Dead Sea (Yam Hamelach) - Israel - Chabad'