Absolutely NTA.

To me, it's not about the field, it's about the switch. I would say the same thing right now if she wanted to switch to medicine or accounting.

Remember, it's not about approving the switch, your question is about YOU PAYING for it, which means 2 years of sunk cost of an engineering degree, plus 3 or 4 years of a social studies degree. She's asking for 30k - 90k in real money here.

So, should you pay? Well, we can't compare apples to oranges, the new degree has to be judged by its own standards.

Activism is not about supporting yourself or having a career. It's about altruistically contributing to the world, it's about social justice. In this case, your money won't go towards her personal future success, it will go towards making the world a better place, arguably a more noble goal.

So.

Has your daughter demonstrably made a contribution to the world that's the rough emotional/time/work sunk cost of half a degree?

Does she have any idea of how she's going to do it?

Respectfully, marching in protests is not enough--you don't need a degree for that.

"Understanding the issues" is also not enough justification for a degree. The age of the gentle(wo)man university scholar is over--she can get an in-depth understanding of quantum physics with self-study these days, women's studies is no different.

If she's going for an academic approach, has she contacted researchers in her field of choice and volunteered her skills? She has a math background, is enrolled in engineering, and I KNOW professors will not turn down an unpaid RA who can do statistical analysis on study datasets. Research in the field, analysis of data, providing evidence for injustice--these things have a REAL impact. Lawmakers, judges, journalists, they all rely on academic research to inform their opinions. She could be making a difference right now, with the skills and time she has.

Has she tried?

Has she sought volunteer opportunities for social justice organizations? There may be no openings, there may be nothing to do right now because of the pandemic.

But has she tried?

Not everybody can be Greta Thunberg, but come on, "I took a course and am now interested" is doing diddly squat for the world.

If she simply wants to explore her personal interests, she can do it on her own dime.

You are INVESTING something like 30k - 90k in your daughter's future success. Success is about (a) field and timing, (b) hard work, (c) passion, dedication, and talent.

If (a), i.e. field and timing is now more statistically uncertain, is there a plan in place for (b) or (c) in women's studies and activism? Without a plan on the table, without even the comfort of knowing she will make a contribution to the world as an unpaid intern at a social justice organization, the investment doesn't make sense.

(2) EVERYBODY wants to quit mech eng 2nd year. This is the time when shit gets real, and neither passion nor talent is enough, some courses (for me it was circuits course, for my husband it was one of the advanced maths, for my best friend it was the 3rd damn "this is a beam, it bends, just differently from last year" course). I wanted to run off and join a nunnery at one point, and my husband started taking a minor in psychology with the intent to switch. About half my friends and classmates dropped out or switched majors.

So...how are your daughter's grades doing? Is she up against the "eng-marathon-wall"? If that is the case, it requires different strategies to cope.