to (one's) name
In one's possession; owned by one. By the time I was out of college, the only things to my name were the clothes on my back and the broken-down old car I had bought in high school. When I stepped off the boat I didn't have a penny to my name, but the future ahead of me felt bright. They used to be so well off, but when the economy crashed and their business closed, it left them with very little to their name.
to one's name
Owned by one, as in He has not got a nickel to his name, or She has only one pair of shoes to her name. This idiom was first recorded in 1876.