When the Book of Genesis (4:26) says that “men began to call upon the name of the Lord,” what does it mean?
The question asks for an explication of Genesis (Chapter 4: Verse 26) from the Bible. The full verse is “and to Seth, to him also was born a son; and he called him Enos: then men began to call upon the name of the LORD.” (KJV; other translations are similar.)
Explication of specific Bible verses always requires an understanding of the context for the verse in question. At a minimum, this means the other Bible verses leading up to that one, as well as verses which the author of that verse may be assumed to have been familiar with which would have informed their understanding in writing the specific verse. Given that the specific (human) author of Genesis is not known, nor the specific geographic or chronological context in which they recorded it, we can only incorporate the context of the rest of Genesis preceding this verse. Ultimately, interpretation can be a highly personal thing.
The story of Genesis up to this point is the story of the initial creation, and Adam and Eve’s fall from direct relationship with God. By Book 4 of Genesis, they have been cast out of Eden, and have begun procreating. Cain has killed his brother Abel, and Adam and Eve have conceived Seth, ostensibly to replace Abel. Seth in turn sires Enos. Prior to this point, there is no record of Adam or his lineage “calling upon the name of the Lord,” which is to ask the Lord for help. More generally, calling upon the name of the Lord could be interpreted as taking the first step in establishing a relationship with God. Therefore, one could interpret this verse as saying that with the birth of Enos, humanity began the process of reconnecting or reestablishing the direct relationship with God that was lost when Adam ate of the forbidden fruit in the garden of Eden.