Do we have some sort of church record from where he came from like we do Shakespeare? Is it just a nickname we gave to him, kinda like how we named one of the oldest human bodies found "Lucy"? Did he leave an engraving of a rock somewhere in the former location of Vinland that says "Lief Erikson wuz here?"
The Norse did not keep church records. Leif's name comes from the Greenlander Saga, which details the travels of Leif Eriksson and names him as such. The Norse sagas usually name the people chronicled, and often provide a short background on their family and relations (probably to make sure the listener knew who they were told about). Like many of the Norse sagas, they were written down from earlier oral traditions by Icelandic monks and priests during the middle ages.
The earliest version of the Greenlander Saga we have is in the Flateyjarbók manuscript, written by two church men from Iceland sometime in the 14th century.
You can read an English translation of the Greenlander Saga here:
https://notendur.hi.is/haukurth/utgafa/greenlanders.html
As you can see from the text, Leif Eriksson is named several times, as are his father Erik the Red and a short geneology is provided.