When you reach the end of your rope, tie a knot in it and hang on.
The earliest citation yet found does not attribute this to Roosevelt, but presents it as a piece of anonymous piece folk-wisdom: "When one reaches the end of his rope, he should tie a knot in it and hang on" ( LIFE magazine (3 April 1919), p. 585 http://hdl.handle.net/2027/wu.89063018576?urlappend=%3Bseq=65).
Misattributed
Variante: When you come to the end of your rope, tie a knot and hang on.