Oscar writes, "I guess due to a mixup in how names are written and understood in other countries and the United States, the names on my passport and I797/I94 do not match perfectly. When I applied for a Social Security number with the Social Security Administration, after a wait of several weeks, I got a letter stating that the DHS was unable to verify the immigration document I submitted as evidence of my lawful alien status. What should I do now?"
I am assuming that you received this legal authorization to be in the United States through an attorney and you should contact the law firm because clearly they did not file the paperwork accurately. You might need to start with your passport. Does the name match that in your birth certificate? If not, go to your own government and get a more accurate passport. Then your attorney can file paperwork with the USCIS requesting a correction of your I797. It is much better to fix this minor mixup at this point because it will produce even more headaches if you choose to apply for permanent status or try to naturalize. Obviously, as soon as the information on your passport matches perfectly with the information in the DHS database, the SSA will issue you a SS#.