How could Abram go to a place called Dan when the person it was named after hadn’t been born yet?

Q.  It says in Genesis, “When Abram heard that his relative had been taken captive, he called out the 318 trained men born in his household and went in pursuit as far as Dan.”  But I thought that Dan the tribe, descended from Dan the son of Jacob, came before the land known as Dan. So how could the land of Dan appear in Genesis before the tribe of Dan?  

The biblical authors typically use contemporary place names within historical accounts, even when the places in those accounts didn’t have those names at the time of the action described.

The city of Dan was actually known as Laish at the time of the events described in the account you read.  It would not be named Dan until several hundred years later, by descendants of Abram’s great-grandson.  But the biblical author uses the name for this place that readers will recognize.

In a couple of places in this same account, the author uses the old name and then gives the contemporary one:  he speaks of “the king of Bela (that is, Zoar)” and “the Valley of Siddim (that is, the Dead Sea Valley).”  But as a rule, the biblical authors use contemporary names.

We see this most clearly in cases where an account actually describes how a place got its name.  For example, we read this in the story of Gideon in the book of Judges:  “They also captured two of the Midianite leaders, Oreb and Zeeb. They killed Oreb at the rock of Oreb, and Zeeb at the winepress of Zeeb.”

What the account obviously means is that during this battle, the Israelites caught up with the Midianite leader Oreb at a certain large rock, where he was killed, and ever since then, that rock has been known as the “rock of Oreb.”  They overtook the leader Zeeb at a certain winepress, where he was killed, so it got the name “winepress of Zeeb.”  But the story is told as if those places already had those names.

It would be just like somebody today talking about something that Kareem Abdul-Jabbar did while playing basketball in college.  At that time, his name was actually Lew Alcindor.  But we use the name that our audience will recognize.

Lew Alcindor (later Kareem Abdul-Jabbar) coached by John Wooden at UCLA