What is the Principal Business Code if I was a university student doing a paid research project

A few possibilities:

541714 Research and Development in Biotechnology (except Nanobiotechnology) 
541715 Research and Development in the Physical, Engineering, and Life Sciences (except Nanotechnology and Biotechnology) 
541720 Research and Development in the Social Sciences and Humanities

Or just use the generic 999999.

A better question is: does a university student doing a paid research project have to report their income as self employment.

The usual answer is  no, depending on the details. 

If this was essentially a stipend, enter at Educational Expenses and Scholarships, under Deductions and credits (not the income section).

After answering no to having a 1098-T*, answer yes to qualifying for an exception (that gets you to the entry screens). You will have to go thru the whole education interview to get to the scholarship screen. At the scholarship screen, enter the amount of the grant. When asked if any was used for room and board, answer yes. Then enter the amount you want to be taxable (usually all of it), in the pop up box. R&B are not "qualified educational  expenses".  So, this is how you tell TT that it is taxable. Note the wording at that screen “or other expenses”. You didn’t have to literally use the scholarship for R&B.  This will put it on line 8r of Schedule 1 (this line was added in 2022).

*If you do have a 1098-T, one of the follow-up questions will be do you have any scholarships not shown on the 1098-T. Enter the additional scholarship/stipend there.  

Scholarships are a hybrid between earned and unearned income. It is earned income for purposes of the $14,600 filing requirement (2024) and the dependent standard deduction calculation (earned income + $450).  It is not earned income for the kiddie tax and other purposes (e.g. EIC).  For grad students and post grad fellows, scholarship, stipend and fellowship income is earned income ("compensation") for IRA contributions.

Taxable scholarship goes on line 8r of Schedule 1, from which TT treats it as hybrid income.