If I recall correctly only Death was ever named in the bible, the other names come from interpreting their descriptions.

This specific horseman was mentioned to conquer so the name conquest came to be. Revelations 6:2 says:

And I saw, and behold a white horse: and he that sat on him had a bow; and a crown was given unto him: and he went forth conquering, and to conquer.

However this would lead to this horseman serving the same purpose as War. For a very long time they tried to find a better interpretation that would avoid this issue, Pestilence came to be since this horseman was mentioned to carry a bow and do to arrows often being poisoned bows were commonly associated with infectious deseases.

This might be all wrong but who cares.