What is the name of this current called in this regulator datasheet?

This is the one:

So typically 5 mA, maximum 8 mA

How to quickly find that: I know it's a current so I look for any parameter specified in A (or mA, uA) in the UNIT column. Then I look at each and see if that could be the one.

There will be a small increase with increasing load current, see the line under the red rectangle.

In that particular data sheet, they seem to be calling it "Input Bias Current". A more common term is "Quiescent Current".

It's called input bias current and is on page 3 of the data sheet you linked.