The USCIS considers Surname/Last Name for the primary identification issue. If you get Blank Surname in your passport and you do not get it fixed, the USCIS and the US consulate will issue your H-1B approval notice and your US Visa with the existing first name in the passport as your Last Name and the first name will be mentioned as FNU. Now, as you have new passport with updated name, you have to get a new I-797 by filling I-824 application. Upon receipt of the new H-1B approval notice with your correct name, you can contact CBP for name correction in your I-94. Typically, it is not required to go for a new US Visa Stamping after this Blank Surname Correction. You will need to carry your both (old and new) passport. Next time when you will go for visa stamping, the US consulate will update your information according to your new passport and approved I-797.
Rani Emandi, Esquire
Emandi Law Firm PC
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