The spelling in "Pursuit of Happyness" is deliberate.

 

The following  is from Yahoo!Answers:

http://answers.yahoo.com/question/index?qid=20081016095141AAk83EC

"The word "happiness" in the title is deliberately misspelled, just as it was on the wall of a day care center where Gardner once sought care for his young son, Chris Jr., during some of his worst days." Chris (Will Smith) even points out the misspelling to the Chinese care-giver that the mural decorating his son's daycare is spelled incorrectly. Chris wanted to make sure that his son was taught things the correct way, so that he could grow up to be an educated man and make something great of himself. Chris "took offense" to the fact that the world happiness was spelled wrong on the wall and became irritated by it. But the word comes to signify Chris' own pursuit of happiness. To quote from the movie:

"It was right then that I started thinking about Thomas Jefferson on the Declaration of Independence and the part about our right to life, liberty, and the pursuit of happiness. And I remember thinking how did he know to put the pursuit part in there? That maybe happiness is something that we can only pursue and maybe we can actually never have it. No matter what. How did he know that?"

http://abcnews.go.com/2020/story?id=2003…