Has there ever been a trend in academia like the one we're experiencing with Data Science? At a panelist event that I attended recently, exactly 100% of the current students and alumni in the room who didn't have PhDs in Machine Learning stated that they wanted a PhD in Machine Learning -- and they all wanted to know how and where to accomplish this goal.  I'm honestly not sure whether most of these students can even describe what machine learning is.  There were long lines to talk with computer science PhDs, who are directors of research either at a university or in industry, e.g., Google, Goldman Sachs - everyone was so infatuated with the thought of getting into Data Science.

Does any field in the past compare to the current field of Data Science, in terms of trendiness and importance?  

Was it Probability Theory?  Fourier Analysis?  Number Theory?  Mechanics?      Law School?

This coming semester, I really wanted to follow my interests and study measure theory and functional analysis - just skimming through the table of contents of the functional analysis book by Reed and Simon was pretty exciting in itself.  Yet, it's hard to drown out the Data Science influence, and I feel that I should be "wiser" and take machine learning / data science courses instead.  

A respected math PhD student in our department advised me that, "if you study some subfield of algebra deeply, then in the future perhaps a dozen people might understand your work.  But if you get into Data Science, it's a field that's projected to grow for decades to come, and if you land in industry instead of academia, all of the skills are transferable, unlike the skills gained from studying algebra."
Does any field in the past compare to the current field of Data Science, in terms of trendiness and importance?

Yes. It has happened before, it is happening now, and will again.  There was programming in the 80's and 90's.  Then Software Engineering... 
Back in the late 40's and 50's there was a big boom in nuclear physics, and nuclear engineering.

Don't get a degree because you think that there might be good jobs.  Do what is interesting to you (and something that is likely to get you a job.)  If you like Data Science, then do it.  If you don't, then don't.  you are going to have to live with doing the work all day, every day for many, many years to get a PhD.  If you aren't in love with it, then you will not succeed.   

Since a PhD is a long-term commitment, takes a lot of patience, costs a lot of money and takes a lot of time during the term, it is not to be undertaken by an answer on the internet. It is your decision.