There are many unhappy rich people in the world, whose extreme wealth has clouded their judgment to such an extent that they cannot maintain even the simplest or most basic of functional human relationships. These people are also generally unscrupulous and unethical in the acquisition of even more wealth.
The possession of great wealth also often leads to health concerns like substance abuse and overeating, which further creates misery for the individual.
In these cases, the statement that "money cannot buy happiness" is very true.
On the other hand, there is no reality within which one could say that those without money are particularly happy. A person experiencing hunger on a daily basis, for example, is definitely not happy. A person who sees his family suffer, without food, transport, education, or clothing, cannot be happy. No amount of love could restore happiness if such a family does not receive some form of help.
In both extreme wealth and poverty, then, money cannot buy happiness.
But it does help.
Money buys gifts, which can make a child happy. The parent who buys the gifts is happy when she sees her children happy. Another famous quote also claims that it is better to "give than to receive." To obtain this blessing, however, there must be something to give in the first place. Hence, a person who wants to experience the blessing of giving needs money to do so in the first place.
Finally, being comfortable, with enough food, clothing, shelter, and so on for oneself and one's family means that one is happy. To be able to meet even one's most basic needs, one needs money. This is why, even though money cannot buy happiness, it can buy things that makes a person happy. While a rich person needs functional interpersonal relationships to be happy, no poor person with such relationships can be happy unless the needs of all those she cares for are met.
One might therefore conclude that money is indeed a great contributor to happiness.