How might having a lot of oil affect Southwest Asia?

This answer assumes that the geographical designation "Southwest Asia" refers to the region encompassing Afghanistan, India, Pakistan, Sri Lanka, Bangladesh, and Nepal. This would be consistent with the U.S. Government application of that phrase. Many reference works, however, use the designation "Southwest Asia" to refer to the portion of the Middle East that includes the Arabian Peninsula (e.g., Saudi Arabia, Kuwait, etc.). The distinction is obviously important in terms of responding to the student's question with any degree of accuracy, but it is also important in terms of the enormous oil wealth already identified and exploited on the Arabian Peninsula and the paucity of oil wealth in the region that includes India, Pakistan, Afghanistan, and the other nations mentioned above.
The impact of the discovery of major oil reserves in Southwest Asia would be largely dependent upon the seriousness with which the governments across that region applied revenues from the sale of oil on international markets to legitimate projects like infrastructure and poverty alleviation. Too often, countries that should be wealthy because of their natural resources are in fact quite poor due to corruption and mismanagement. Nigeria and the Democratic Republic of Congo are both blessed with enormous natural resources including oil and minerals. Corruption and conflict, however, have precluded the levels of affluence among their respective populations that the sale of such resources should have provided. Similarly, Venezuela has huge oil resources but is experiencing a great deal of internal turbulence solely because of mismanagement on the part of the government.
In contrast, oil-rich nations like Kuwait and Saudi Arabia have been able to develop economically because of their use of oil revenues for domestic projects (although corruption in Saudi Arabia has limited the benefits that could have accrued from its oil industry). In the case of Southwest Asia, corruption, mismanagement, and internal strife all exist in abundance. All these countries could benefit enormously from the discovery of major oil deposits, but endemic corruption in Afghanistan and Pakistan has already taken a serious toll on those countries’ economic development. Sri Lanka profits from its exports of spices and tea but wasted tremendous wealth to a decades-old conflict between its majority Sinhalese and minority Tamil populations. Suffice to say, however, that revenue from oil could be used to develop infrastructure, especially in India, which has a large economy but millions of people living in poverty without basics like plumbing, and in the notoriously ungovernable Pakistani Northwest Frontier Province and the Federally Administered Tribal Areas, which have served as havens for terrorist groups like al Qaeda because of their central governments' inability to manage those regions. Afghanistan possesses a great deal of potential wealth courtesy of its sizable deposits of “rare earth metals,” but internal corruption and strife has prevented the efficient exploitation of those deposits.
In conclusion, oil wealth can be used in Southwest Asia to develop more efficient infrastructures and healthcare systems and to alleviate poverty. Whether such wealth would be applied to such programs, however, is unclear.