Is it unwise to pick an old field of research to write a PhD thesis about?
If you pick an overly-trendy area there is a chance that it will be easier to get a Ph.D. in that area but that many mathematicians will be less than impressed by the resulting degree. For example, "fuzzy" is (or has been?) a trendy concept. There is almost a template for getting a Ph.D. in it (albeit one which is getting harder to apply by sheer competition): pick some topic in pure mathematics for which there doesn't yet exist a fuzzy version (for an example the exact phrase "fuzzy symmetry group" has only 4 Google hits). Learn the classical theory. Fuzzify it: whenever you see the word "set" replace it with "fuzzy set". Prove something -- which with lack of competition shouldn't be all that hard. Anticipating the question as to why anyone should care, hand-wave about potential applications to artificial intelligence. Voila! You have a dissertation (I am exaggerating a bit, this is of course easier said than done, and a successful fuzzification will involve more than simply replacing sets by fuzzy sets).
Now I don't deny that the core concepts of fuzzy logic and fuzzy set theory have a great deal of mathematical depth, and that much of the fuzzification of classical pure mathematics has been well-motivated. Still, it seems likely that much of it has been done just because it was low-hanging fruit for somebody who wanted to write a dissertation or paper. I am instinctually skeptical of the value of a paper when it contains the word "fuzzy" in the title. I won't dogmatically reject it, but I will need some convincing as to its value. If your dissertation has a trendy buzz-word but otherwise seems unmotivated, you might meet with similar skepticism.
All of the areas you mention are extremely broad areas and extremely popular. For instance, it doesn't make sense to call number theory a trendy topic or not, except in the sense that there are many trends that come and go within number theory. Moreover, being "trendy" is essentially independent of being "relatively easy." Two topics that are sort of trendy in number theory close to me now are p-adic Langlands and beyond endoscopy, both of which are incredibly technical and require a huge amount of background to get into, whereas some other trendy parts of number theory like Apollonian circle packings and Ramanujan graphs are much easier to get into.
Just because a certain subject is more established doesn't mean it's harder to work in. It just means it's richer and has more subareas to specialize in.
What you should do your thesis is an area that (1) you would be happy working in, and (2) you can find a suitable advisor for. In any case, easy should not be a reason for doing a PhD (most people will tell you it is not easy---mine turned out to be, but I got lucky).
Note: there are certain areas that have fallen out of favor, and it can be harder to get an academic research position if you work in one of these areas, unless you can connect it to things people are interested in nowadays. However, there's a big difference between not being trendy and being comatose.
I'm going to say some stuff based on my own Physics PhD. Disclaimer: These are just my opinions and are undoubtedly affected by my cynicism.
PhDs are not all created equal
I'm of the belief that PhDs are not equally difficult. Additionally, PhDs are not equally valuable. In fact, value does not necessarily correlate with difficulty. I knew students who were given projects that turned out to be gifts - a simple idea with simple execution that produced papers like a gold mine. Some PhD projects are absolutely easier than others so don't let anyone persuade you otherwise.
Myself, I got landed with something that was borderline impossible (professors from other labs expressed their sympathies when I described my work) and, though I eventually succeeded, trying to get scientific value out of it was like trying to get blood out of a stone.
Here's my advice:
- Comprise a list of all the current 'sexy' on-trend topics in your field
- Choose the one that most genuinely interests you
That's it. My justification is as follows:
- Point 1 increases your ceiling for success (incremental improvements to old methods won't skyrocket your career), point 2 increases your chances for success (because you'll remain enthusiastic and see it through when the going gets tough)
- On-trend subjects get a lot of attention at conferences. Attention leads to corporate and government interest, which in turn leads to grants. I've got to tell you that, however much you enjoy your PhD project, you'll enjoy it a lot more with increased funding.
- On-trend subjects produce material that'll make your posters and presentations attract other academics like bees to honey. Networking is absolutely vital in academia and choosing a on-trend subject will net you a lot of powerful friends.
- You might have an easier time getting published if your research field is truly cutting edge. I published several papers detailing improvements to existing scientific methods, but I don't mind telling you that it was an uphill struggle to sell the idea to the reviewers.
- I don't see the problem in choosing an on-trend subject to make your life a significantly easier, providing that you're actually interested in that subject
- If your field actually takes off, your name will be among those few who pioneered the research. You don't need me to tell you how valuable and fulfilling that would be.
Hope that helps.