Excellent question! And one that is impossible to answer in just a few short lines ... but I will try. David purchased the threshing floor of Arunah (also spelled Aranuah in 2 Samuel 24 and Ornan in 1 Chronicles 21) in order that he might make sacrifices to God and stop the plague that had killed 70,000 men in one day. A plague caused by David's sinful act of conducting a census that was not ordered by God.
If you go back and read the passages in 2 Sam. and 1 Chron., you will see that the Angel of the Lord was stopped in the conducting of judgment at basically the "front door" of Arunah's property (2 Sam. 24:16). David then went to Arunah's home to build an altar to God in thankfulness for God's sparing of the people. David went to buy and Arunah attempted to give. David refused to offer a sacrifice to God that cost him nothing and then purchased the threshing floor from Arunah for six hundred shekels of gold.
The next time we see mention of this threshing floor is in 2 Chron. 3:1. Solomon begins the construction of the Temple on the site where David purchased the land so that he might seek forgiveness from God for his sin. The site was purchased by David, king of Israel, and therefore one could assume that the land therefore belongs to the descendants of David - the Jewish people.
And only if the answer was that simple - ownership determining property rights. However, it is not that simple because the land and the spot have been subjected to many rulers since AD 70. However, the final question which must be asked and answered is to whom did God intend for it to belong? For that answer one can refer to not only 2 Samuel 24 and 1 Chronicles 21, one can also go back to Genesis 12, 15, 17, and 22. Genesis 12, 15, and 17 refer to the covenant established between God and Abraham regarding the land and his descendants through Isaac. Genesis 22 refers to the binding of Isaac (an archetype I believe of Jesus' sacrifice). And where did this event in chapter 22 take place, somewhere on Mount Moriah or the place where King David purchased from Arunah a place for sacrifice and a place which would know be known as the Temple Mount.