The phrase "the name of each" is grammatically correct and can be used in written English.
Example: The teacher asked for the name of each student before starting the lesson.
The name of each starter was heckled with collective booing.
Lift it, and the name of each year's winner is engraved inside.
As the name of each child is called, their parents run and dance towards them.
The only sound was the name of each murdered child coming through loudspeakers.
Portable blackboards list the name of each beer and the date when the keg was tapped.
Introducing the survivors, actress Angela Bassett read the name of each of the victims.
A hand-drawn apple appears by the name of each candidate.
Snyder let her buy it only after she had memorized the name of each one.
Separate plates bear the name of each high school and the signature of its principal.
The weeklong process came down to one word, circled by the name of each person who was chosen: Si.
The bleacher fans belted their usual roll call, chanting the name of each player on the field.
