He can engage in insider trading.
Now, you may say, insider trading is illegal. But actually this is not entirely true for politicians. For instance, it was not illegal for members of Congress to sell or hedge their portfolios before they passed TARP in 2008.
Furthermore, while you say the POTUS has to spend most of his time doing his job and doesn't have time to actively play the market, it doesn't really require much time to transact a trade here and there. There are plenty of mobile apps out there for various brokerages which will even allow you to initiate a buy or sell order from anywhere you have cell coverage. You can spend less than 5 minutes a day doing it, say during a restroom break before you make a speech.
Now, how will this work? As President of the United States, you have a great deal of inside information not only on what laws are likely to make it to you, and input on what laws you want to reach your desk, but veto power over laws, and even executive orders which bypass the need for legislation. Many of your acts as President can radically change the value of various publicly traded companies. Making a billion dollars is quite doable using highly leveraged instruments such as options and futures - things which normally would have very high risk if not for your inside knowledge of how your policies and legislation would proceed.
And you don't even need to compromise your ethics: for instance, you could buy cheap out of the money puts in a certain large oil company having significant interests in a remote Alaskan oil field (betting the price will fall) and then while nobody is expecting it, sign an executive order banning drilling in this region due to the environment impact it will have on native wildlife. Conservation organizations will praise you. Or you can position your money so you will make a killing when you announce your new health care initiative.
And you don't even have to limit yourself to local concerns. 207
The actions of the President of the United States can even affect the value of the dollar. Invest heavily betting the USD will fall, and then announce policies to be taken to devalue the dollar to increase the competitiveness of American made goods abroad. (Unions love you) Or vice versa if you feel imported goods should be cheaper and benefit from a stronger dollar (consumers love you). Your views are not important to achieving your goal, whatever you believe in and whatever policies flow from that can facilitate your making a great amount of money. It doesn't really even matter if your goals are ultimately unsuccessful, because you can cash out after your initial announcement moves markets in the desired direction.
While you can make your positions strategic to benefit a large part of the population which will in turn re-elect you, you probably won't even actually need to be re-elected to a second turn. The combination of geometric returns on your initial investment combined with the power of what you say to move markets should take much less than 8 years. For a large part, your salary is not even a big part of this. As an illustration of leverage, in under two years, <PERSON> turns \$1000 into nearly \$100,000 in cattle futures. You are going for these kinds of moves, but overnight. The fact that it takes you a year take make $400k will pale in comparison. 207