What were the two prized possessions of Della and Jim?

In a more conceptual way, Jim and Della's most prized possessions are one another and the love they feel for each other. While love is not a tangible possession, like a watch or one's hair, it is clearly of greater value to the young couple because they are willing to sacrifice their material possessions in order to buy thoughtful gifts for each other; the reason they are so anxious and desirous to buy thoughtful gifts for one another is because of their shared love. Near the end of the story, the narrator says, 

But let me speak a last word to the wise of these days: Of all who give gifts, these two were the most wise. Of all who give and receive gifts, such as they are the most wise. Everywhere they are the wise ones. They are the magi.

Jim and Della are the most wise, even more so than the wise men who attended Christ's birth, because they realize that their own sacrifice—sacrifices they were willing to make out of love—is ultimately more valuable than anything they might own. In this sense, then, you could even argue that the gifts they buy for one another are their most valuable possessions—not because they have are worth a great deal of money, but because they are symbolic of the love Jim and Della share and their willingness to sacrifice for one another.