Having run of success, one's name honourably entered here? (2,1,4)
I believe the answer is:
on a roll
'having run of success' is the definition.
(on a roll means having a string of successes)
I don't understand the rest of the clue.
(Other definitions for on a roll that I've seen before include "Having long spell of good luck" , "Hitting a purple patch (as bacon might be?)" , "enjoying continuing success" , "Where toilet paper may be found" , "Having a run of good luck (colloq.)" .)