Walking Eagle
One report claims that Native Americans held a special ceremony and honored President George W. Bush with the native appellation, “Walking Eagle”:

President Bush was invited to address a major gathering of the American Indian Nation last weekend in Arizona. He spoke for almost an hour on his future plans for increasing every Native American's present standard of living. He referred to his career as Governor of Texas, how he had signed "YES" 1,237 times, - for every Indian issue that came to his desk for approval. Although the President was vague on the details of his plan, he seemed most enthusiastic about his future ideas for helping his "red brothers".

At the conclusion of his speech, the Tribes presented the President with a plaque inscribed with his new Indian name - Walking Eagle. The proud President then departed in his motorcade, waving to the crowds. A news reporter later inquired to the group of chiefs of how they come to select the new name given to the President. They explained that Walking Eagle is the name given to a bird so full of shit it can no longer fly.

Before you get too gleeful over this one, let me tell you that it is a hoax that started circulating on the Web. Actually, there was also a similar version that circulated about John Kerry during his 2004 presidential candidacy, but instead of “Walking Eagle," he was dubbed “Running Eagle.” Same punch line, neither can fly because of the same overload.

Targets for Jokes
Presidents are prime targets for jokes. There is a site dedicated solely to George Bush jokes, including links to jokes by David Letterman, Jay Leno, and Conan O’Brien. A word of warning about this site however: if you click on President Bush's 'Quotes Hall of Fame' famous quotes, you will be assaulted by a list of things Bush never said.

Most statements on the list were malapropisms that were either said by Dan Quayle or claimed to have been said by Dan Quayle. For example, the first on the list "If we don't succeed, we run the risk of failure." This one was attributed to Dan Quayle by humor writers merely putting forth what they think Quayle might say.

Misunderestimated
Good-natured humor that springs from the real flaws of political candidates is all fun and harmless. In Bush’s case his sometimes less than stellar use of language is fair game; for example, his conflating the words “misunderstand” and “underestimate” which resulted in “misunderestimated.”

However, in Bush’s case when the less than good-natured bantering becomes downright dirty and inaccurate, the dirt slingers are just demonstrating again their “misunderestimating” the abilities of this confident, capable president.

President Bush is self-deprecating; he does not mind making fun of his own foibles. He says about his wife that she speaks better English than he does. But this president is so solidly grounded in soul qualities that he can take all the ribbing, and he knows that the vicious inaccuracies will fall away as all inaccuracies do.

References:
Running Eagle from Snopes also includes the “Walking Eagle” story
GWJokes.com site featuring jokes about the president
Quayle Quotes from Snopes