I just read the following text on tolkiengateway.net:
The name Moria means "Black Chasm" and was a derogatory description of the place which the Dwarves did not like, and was given after Durin's Bane took over the city in the Third Age. It is therefore a mystery why that name appears on an inscription made in the Second Age, and made in consent with the Dwarves.
It discusses the inconsistency of the name Moria appearing on the Doors of Durin. Was Tolkien aware of this? Did he ever comment on this?
I am not asking why the Dwarves dislike the name, but if they could see the future? How can the name Moria appear on the doors, which were built in the Second Age, when it was given in the Third Age?
A rare friendship sprang up between the Dwarves and the Elves of this new land. Eregion's ruler, Celebrimbor, helped to construct the famous and magical gate that became known as the West-gate of Moria..
In that time it was given a new name, Moria, the Black Pit.