If a certain person goes from India to US, earns billions of dollars there, and brings the money back to India then is he contributing to the improvement of the economy of India (doesn't matter if the contribution is negligible)?

Yes he certainly is, but he can very well contribute the same while earning in India, it depends on the type of work you are doing and what does it fetch in the global market. Guess Kumar (An educated person), goes and works in USA for 10 years  accumulating a wealth of 1 million dollars. These one million dollars are the promise of American Monetary system of paying back Kumar by equal amount in goods and services. During his stay in America Kumar must have added more than 1 million dollars as goods or services to American economy, say $2M, unless he was involved in some illegal activity. So once he comes back to India he is only bringing with him $1M, while he could have contributed far greater to Indian economy had he worked with an equal number of opportunities and resources. So it depends upon what an individual can contribute to an economy, home or abroad. If the total benefit that is carried back home ($1M) is greater than the total benefit that could be made by staying at home, It is very well worth it to work abroad, more so if you are unemployed/underemployed in India. But if the contribution carried back home is insignificant in comparison to the benefit that could be generated by staying at home, better stay home.

This is what precisely is happening now. India is the largest recipient of expatriates remittances of about $80 b per annum. This to some extent bridges the gap between exports and imports. That is whyNRIs are wooed by the government.  Whenever Indian rupee depreciates one of the first things done by RBI is to increase the rate of interest forNRIs for their FCNR deposits.

Since you can't use dollars in India, you'll have to convert it to Rupee. When you sell your dollars to a bank it will be added to RBIs foreign reserves. So yes, he would be contributing to the economy and helping Rupee gain against dollar. But this is totally theoretical. Bringing in billions, or even millions for that matter, is no joke. LN Mittal made a fortune and he did his college in India. So he is setting up a steel plant in India. That's FDI. That's the way money comes in in real life. But it's a two way process. Foreigners will also come to India, set up their businesses and walk away with the money. For a country to prosper, what it really needs is to increase its own productivity. Industrial outputs, manufacturing, agriculture all need to be increased domestically to become competitive at the global level.