What is the relativistic four-vector version of ordinary velocity called?

The relativistic four-vector version of ordinary velocity is called four-velocity.

In special relativity, ordinary velocity describes how an object's three spatial coordinates change with coordinate time. Four-velocity instead describes how the object's four-position changes with its own proper time, the time measured by a clock traveling with it. It is written as U = γ(c, v), where v is the ordinary three-velocity and γ is the Lorentz factor.

Four-velocity belongs to the geometry of an object's world line through spacetime. Its Minkowski magnitude is an invariant with magnitude c, although the sign depends on the metric convention. For an object at rest, its spatial components vanish and only the time component remains.

A common mix-up is four-momentum: multiplying four-velocity by invariant rest mass produces four-momentum. Four-current similarly combines charge density with four-velocity, while the four-gradient is a differential operator, not a velocity. Four-velocity is defined for massive, timelike objects; light has no proper time and therefore no ordinary four-velocity.