Everyone's answer to this question will be different. Just as many of the items carried by the soldiers in The Things They Carried are unique in some way and give the reader insight into the characters as individuals, what you choose should speak to who you are in some way.
If you are having trouble thinking of what item to choose, imagine that you are going away for a year or more (just like the men in the book) and can only take a few small personal items with you. Perhaps this item has a sentimental association, like Dobbins' girlfriend's pantyhose or the Bible that Kiowa carries which was given to him by his father. Do you have a gift from a loved one that has special meaning to you? Perhaps that could be your item. Think about why it is meaningful, what it says about your values, and the importance of the relationship you have with the person who gave it to you.
Perhaps you have an item that speaks directly to a way that you identify. This could be jewelry or a special piece of clothing. Did you buy it for yourself? If so, why? What attracted you to it in the first place? Is it something that you only wear or use on special occasions or is it an everyday item?
Whatever item you choose, there should be a story behind it. This story should include how it first came into your possession, why it has special meaning to you, and what this says about you as a person.