I think what is meant by this old cliche is that the things in life that will grant you lasting happiness - love, friendships, people etc - can't be purchased with money. Granted, this is probably true. What it has turned into, however, is people saying "Money will not make you happy" and that is not true. At all.
This has become accompanied by other common attitudes - the primary one I can think of being that money corrupts people - and has made it so that having money is almost stigmatised in society. It's a recognised part of society that the rich are hated for being rich. That's why I don't believe in the phrase "money doesn't buy happiness". It's become a way for people to console themselves about not having as much money as they'd like to have.