Can an illegal immigrant legally change their first name?

That is possible. The law about name changes in Texas does not require legal immigration status.

A person who is not in a legal immigrant status can petition a state court for a change of his name. The court will, most likely, grant him permission to assume a different name, usually even without asking why does he want to assume that particular name. It is also likely that the judge will require the applicant to notify the Immigration Service about the change of his name. In any immigration proceeding, this person will still have to show his passport and his birth certificate. Even if he can convince the consulate of his country to issue him a new passport with the new name, his birth certificate will not be changed. So, he will have to explain to the Immigration Service (or to the Immigration Court) the story with his names. Actually, it happens often that an immigrant (legal or illegal) uses a name that is different from the name he was given at birth, simply because that's the nickname all his friends call him, or because his real name is difficult for Americans to pronounce, or for any other reason. Immigration Service and Immigration Court are well familiar with these situations and do not mind too much people's using different names (unless one changed his name to hide a criminal past, or a history of immigration law violations, or running away from child support payments, etc.). If the applicant wants to prove that he worked and paid taxes for many years, he would simply have to prove that he is, in fact, the person who did that, although under a different name. Getting a state court's decree permitting a legal name change will not give him such proof.

Anyone can change their name in State court.

A name change does not require a check on immigration status. He can go to the county court house and request that a name change by ordered.

To change the name without legal status (even if it could be done) may give the appearance of fraud. You should fix your status first.