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.