We know some things that cause happiness: Leisure time, successful marriages, giving to charity, etc. And we know some things that cause unhappiness: Unemployment, poverty, etc. It stands to reason that the government could affect at least some of these things through policy...for example, maybe they could help people have successful marriages by subsidizing or encouraging marriage counseling. Maybe they could encourage people to give to charity by giving people tax credits (instead of deductions) for charitable contributions. Maybe they could alleviate unemployment by abolishing the minimum wage, since a person's income doesn't affect happiness as much as having a job does. Maybe they could alleviate poverty through more anti-poverty programs and/or education reforms. Maybe they could affect leisure time (and unemployment) by offering tax breaks to businesses that have a 35-hour work week instead of a 40-hour work week.
These are just some ideas. Maybe not all of them would have the desired effect, but I definitely think that since we know the variables that cause happiness, and that since the government can affect those variables, that therefore the government can affect happiness.