Its going to vary based on your wealth. Likely you'll carry the tools of your trade especially if you're of a slightly lower class. If you're wealthy you might have an object like a signet ring and you might carry a Pugio or Parazonium(swords were banned within the walls of rome itself for much of rome's history). While walking around at night in your home you might carry a oil lamp. When outside your home you could carry a purse, probably made of leather. If you're educated you might own a wax tablet (paper wouldn't be common till later and Papyrus was reasonably expensive so it was easier to use the reusable wax tablets.)

As for things you might pass on the obvious things that come to mind if you are wealthy enough to own them would be a house, land, slaves, animals, perhaps an antique such as an old Greek pot, a weapon, a ship, a cart, a business. One thing that has little equivalent in the modern world however is your family Lares and shrine. The Lares are almost a kind of family god, sometimes they would be like a being believed to watch over the family or house, sometimes they would be the spirit of the family's founding figure, especially if that figure was part god or a legendary or semi-legendary historical figure. Claiming to be a descendent of a legendary figure or deity was quite common, particularly among the upper classes.