Happiness is fleeting. Though I enjoyed everyone's answers, you'll notice that happiness isn't mentioned very often in the bible. Actually, a Biblegateway search will show you that in the NIV it only appears 21 times.
However, what DOES appear a lot in the Bible is Joy.
Joy and happiness are not the same thing...though today's secular culture often uses them interchangeably.
Noah Webster originally defined 'Joy' as:
1. The passion or emotion excited by the acquisition or expectation of good; that excitement of pleasurable feelings which is caused by success, good fortune,the gratification of desire or some good possessed, or by a rational prospect of possessing what we love or desire; gladness; exultation; exhilaration of spirits.
Joy is a delight of the mind, from the consideration of the present or assured approaching possession of a good.
The thing is...happiness is just a feeling we have at times, but Joy is an inner calm and contentment with whatever life has to offer.
11 I am not saying this because I am in need, for I have learned to be content whatever the circumstances. 12 I know what it is to be in need, and I know what it is to have plenty. I have learned the secret of being content in any and every situation, whether well fed or hungry, whether living in plenty or in want. 13 I can do all this through him who gives me strength. -Philippians 4:11-13
Joy is not fleeting, but an internal attitude based in and from Christ.
10 If you obey my commands, you will remain in my love, just as I have obeyed my Father’s commands and remain in his love. 11 I have told you this so that my joy may be in you and that your joy may be complete. 12 My command is this: Love each other as I have loved you. -John 15:10-12
I'm not trying to give you a cliche, or the distant feeling you mentioned by this answer.
What I am saying, as the writer of Ecclesiastes did, is that all meaning, purpose, hope, joy, ext...are found in God alone, and apart from Him, nothing matters. Seriously.
He is not only the Creator of all things, but the Sustainer of them too. The only way I have ever been able to overcome anything is with and through Christ.