Invest
In order to invest in a region your family must have more than 2000 gold of family wealth. Investing causes the region's production to rise dramatically, which causes the region to produce more gold and food than usual. This is what your advisor is referring to when he tells you the expected return on investment, and it's distributed according to the realm's normal tax distribution scheme. Your family will not directly receive anything from their investment.
You will receive one Fame point the first time your family makes an investment.