Graduate student stipends for which the student receives a W-2 Form from the University are usually paid as salaries for employment as a research assistant or teaching assistant or graduate assistant with the latter two title categories requiring the assistant to provide specific services to the University such as teaching a class or a laboratory course or grading homework, holding office hours for students in a course (teaching assistants) or doing programming, computer maintenance, etc (graduate assistants). In contrast, research assistants are generally not required to perform specific services for the University but to assist the advisor in carrying out the research proposed in the statement of work of the research grant or research contract that is paying the assistant's salary. Such grants and contracts are technically awarded to the University, but it is the professor who as submitted the proposal to the awarding agency through the University and it is the professor who decides whom to hire as a research assistant. Often, assistantships include a waiver of tuition and fees whose value is not included in the wages reported on the W-2 Form; it is a nontaxable perk. So, the OP should find out what his status is with respect to his University's tuition and fees. Are they still being waived i.e. does he still have a tuition and fee waiver (which many departments still continue to provide for some students who are no longer research or teaching or graduate assistants -- it provides some financial support for the graduate students in the department who are otherwise unsupported), or is he expected to pay them if not now, then starting next semester?
With regard to how to report the stipend for which the OP has received a 1099-MISC, questions will definitely be asked by the IRS if the OP does not report this income on a Schedule C, if not when the tax return is filed and "accepted", certainly when the IRS gets around to reconciling the information that the IRS has (it gets a copy of the 1099-MISC Form) with the tax return it should have been reported on. Yes, as @chepner points out in his comment on the OP's question, the OP would be liable for paying self-employment tax, both the employer's share and employee's share of Social Security and Medicare tax) on the income from the stipend payment. On the other hand, the OP may be able to deduct the cost of books and office supplies etc needed for his research on the Schedule C. The alternative of trying to get the advisor's new university to treat the stipend as a wage paid to you seems more difficult to accomplish.
