Personally, I think this will be extremely difficult to do with nothing but time to invest. The problem is that just about everything associated with earning money has a cost. (As the old adage goes, it takes money to make money.) For instance, even if I said I'd pay you $2000/month to write online articles for me, you'd still have to make the investment in a computer, pay for internet service, etc.
That said, if we assume that you have some basic resources (a computer, internet access, etc.), there are some things you can do to try to generate income. For instance, there's a site called WebAnswers where you can earn $$$ for asking/answering questions, and you can start earning there pretty quickly (in just a few days). The problem is that you have to remain fairly active there in order to keep earning. Still, I was able to earn about $50/month there back when I was active by answering 5-10 questions per day.
You can also try your hand at self-publishing. I've had good luck in that department and found it to be very lucrative. You can do everything yourself - writing the book, editing it, formatting, etc. - but the people who seem to experience the most success self-publishing are those who pay for expert help in those areas.
If you have an area of expertise you can start offering your services on places like Fiverr. For instance, if you're a good singer, maybe you can offer to write an original song for someone's birthday, deliver a singing telegram, and so on.
In short, there are a lot of things you can do to earn money by only investing time, but in my opinion time is a commodity that people place too little value on. Once it's gone, you can never get it back; it has a worth that you shouldn't trivialize. With that in mind, I would argue that you invest some $$$ into trying to earn passive income. For instance, investing for dividend income is a way to start getting some cash, and there are companies that have been increasing their dividends annually for 25 to 50 years or more. You can buy a decent vending machine for a couple of hundred bucks and get started that way (although it's not truly passive income since you have to restock the machine occasionally).
As you can probably tell, earning passive/residual income is a passion of mine. I think you'll be able to reach your goal, but you'll get there faster if you're willing to invest more than just time (which is pretty precious in my opinion).