Student or Learner
What does it mean when somebody says: I had been the eye of a travelling rat, natural tinker.
--- I didn't find the meaning of "the eye of ......"?
Have you been reading Bono? Those are his words, and he appears to be quoting the poem "The More a Man Has the More a Man Wants" by Paul Muldoon.
This is not a standard idiom that I know of. If the words make no sense the way either Bono or Muldoon use them, here is an analysis that may help.
If I may suggest, consider the beady eyes of a rat as it survives in city garbage and sewers, exterminated but never fully by man. But follow your own imagination!