The easiest way to get into a Ph.D. program is to have a professor who knows who you are and wants to have you as a student. In many other questions on this site, people talk about the difficulty of standing out from the crowd amongst the large number of applicants to any decent Ph.D. program. With this professor, at least, you have stood out, and that's quite significant... if you want to do a Ph.D. with them.
First a sine qua non: don't even consider it if the program doesn't guarantee some degree of funding. Every respectable university should be offering support for its STEM Ph.D. students through some mix of research and teaching assistantships. It should be the department and not the professor that guarantees funding (though if the
professor has funding, they may be able to give you more time as a research assistant rather than a TA, if that is what you both want).
Beyond that, key things to ask yourself:
- Do you actually want a Ph.D.? Getting a Ph.D. is a terrible, soul-wrenching process and it renders you unfit for most employment. If you are truly drawn to research, however, it is the best and only course to take.
- Are you OK with not being at a famous institution? There are only a few famous institutions in any field, and getting into them is a gamble. Often, however, there are a great many solidly respectable institutions that can launch you on a totally reasonable career, especially since it is often possible to "upgrade" institutions through postdocs.
- Do you really want to work with this professor? You don't know what working with this professor will actually be like. One advantage of the high-profile institutions is that if you find things aren't working with your advisor, there are often many opportunities to switch to others. In a smaller and less renowned department, you will likely have less alternatives should things go wrong.
Ideally, if you answer yes to all of these questions, you should start doing some work together now, so that you can get a sense of whether there's a good working relationship. If so, embrace it and be grateful that you found a good match.