wow......that is a strange problem.

by "clicks" i meant that when you turn your key does something under the hood click, that is what low voltage or bad connection or bad solenoid would do, but you already answered that question.

since it eventually turns over normally i would agree with the rest and say you need to test that starter, but just for shits and giggles you might clean all the connections and look for any loose or damaged wires down there while you have it jacked up and before you remove the starter to see if it helps at all.....

just in case, because you never want to do more work than needed