Changing your name won't affect your immigration status. It may be more or less of a hassle if you do it before or after getting your green card. My guess is that the cost and hassle of replacing the green card would argue in favor of changing your name now, before you apply for the green card.
You'll normally change your name through a state court where you reside, though, and then submit the court order with your subsequent submissions to USCIS, rather than effecting the change of name through the green card application itself. You'll probably also want to check whether this has legal effect in your country of citizenship.
Detailed questions on practical aspects of the change belong on Expatriates, however. As far as this site is concerned, the answer is yes, it's fine.