Hang On Meaning

(idiomatic) To persevere.

Example:   Hang on. Let me check.‎
To hold, grasp, or grip.
  Hang on to the handle so you don't drop it.‎
1913, Joseph C. Lincoln, chapter 7, Mr. Pratt's Patients:
  Old Applegate, in the stern, just set and looked at me, and Lord James, amidship, waved both arms and kept hollering for help. I took a couple of everlasting big strokes and managed to grab hold of the skiff's rail, close to the stern. Then, for a jiffy, I hung on and fought for breath.
  Hang on to my jacket until I get back.‎
  The audience hangs on his every word.‎
  He's got a philosophy he hangs on to.‎
  Just hang on and keep going; this pain won't last forever.‎
1972 : Perfect Day by Lou Reed
  It's such a perfect day, I'm glad I spend it with you
  Such a perfect day you just keep me hanging on
  You just keep me hanging on