Jehovah's people have always been considered special and precious to Him. For centuries it was the descendants of Jacob (Israel). They were special by way of being the children of Abraham. They were told they were a special possession, a royal people. After the Jews rejected Jesus, and he subsequently rejected them as God’s chosen people, Peter applied the words previously reserved for the Jews to the early Christian congregation (the whole of Christianity as it existed back then).
Now, this is not an attempt to put forward any sort of official interpretation of scripture, but rather to demonstrate the possibilities. So, In what sense are God's people his special possession? Mal.3:10 invites those serving him to ‘test him out’ to see if he won't keep his word to bless them if they do their part to be loyal and obedient. That is part of what makes his people a special possession--their willingness to accept him. While he makes the sun rise on both righteous and unrighteous (Matthew 5:45), to those loyal to him, willing to acknowledge not only his existence, but his right to tell them what to do, that is "special" to him.
But as a group, believers are also special in that they would be the stellar example of what it means to have a relationship with the Creator. Although the Jews did not preach in the same sense that Christians do, while/when loyal, their loyalty to God was demonstration for those around them that God was truly with them.
Is this such a difficult concept to grasp? It shouldn’t be. Many people speak of others that are close to them as “mine.” “My wife (or husband),” “my best friend,” and other expressions are all common today. Even though other people are not owned (as if slaves), the bond is so close that there is a mutual exclusivity in their minds that each other belong to one another and no one else shares that special one-on-one bond.