Firstly, it was the title of the original short story written by Arthur Conan Doyle. As you may or may not know, contrary to what most modern adaptions of Sherlock Holmes show, Moriarty only ever appeared in a single Sherlock Holmes story, called "The Final Problem".
Arthur Conan Doyle wanted to kill off his detective with this story (more info can be found here: how-sir-arthur-conan-doyle-tried-and-failed-to-kill-sherlock-holmes.
Effectively, the Final Problem was just that - the final problem Sherlock Holmes had to deal with. Moriarty was invented as a mastermind so brilliant that he matched Holmes' intelligence. Their final showdown took place at the Reichenbach Falls in Switzerland, with both characters dying (until Conan Doyle bowed to public pressure and brought Holmes back).
Therefore, with that original story in mind, it seems obvious the tv series wanted to reference it as much as possible. However, given your question is specifically about the tv series, you might be interested in this discussion over at the Sherlock TV Series page: What is the "final problem", which is full of opinions discussing exactly what you've just asked. A particularly interesting interpretation is provided by one of the users, who cites the following two reasons as being his understanding of the "Final Problem":
1. Staying alive without dying of boredom
The tea party at 221b in TRF isn't the first conversation between Sherlock and Moriarty on this
subject. Several phone calls and remarks in TGG (though through a
"stolen" voice from Moriarty) about the puzzles hint at this:
LESTRADE: Why would anyone do this? SHERLOCK: Oh…I can't be the only
person in the world that gets bored.
VOICE OF HOSTAGE AT PICCADILLY CIRCUS: This is about you and
me…because I'm bored…we were made for each other, Sherlock.
VOICE OF OLD LADY HOSTAGE: You are enjoying this, aren't you?
JOHN: So why is he doing this, then? Playing this game with you. Do
you think he wants to be caught? SHERLOCK: I think he wants to be
distracted. JOHN: Oh…I hope you'll be very happy together.
2. Staying number one (Moriarty more than Sherlock:"…you should see me in a crown!"/ Sherlock's deerstalker isn't exactly a crown… )
JIM: Here we are at last – you and me, Sherlock, and our problem – the
final problem. Stayin’ alive! It’s so boring, isn’t it? …It’s just ...
staying… All my life I’ve been searching for distractions. You were
the best distraction and now I don’t even have you. Because I’ve
beaten you.): It was easy. Now I’ve got to go back to playing with the
ordinary people. And it turns out you’re ordinary just like all of
Earlier, at the pool scene, Jim isn't quite sure yet whether to
destroy Sherlock (and John) right away or just send them "a friendly
warning" not to be in his way any longer. He admits that he has
enjoyed "this little game of ours". Fortunately the ringtone postpones
his decision: "Wrong day to die…!"
Jim knows that killing Sherlock will destroy the basis for an
interesting life in the future. A boring life for him isn't
worthwhile. But it is also not acceptable for him to be beaten.
Solving these two problems at the same time is his dilemma. The only
way out he sees for himself is his suicide. The only satisfaction he
has is that he is convinced that has destroyed Sherlock's reputation
and life as well.
Whilst you are free to peruse the link to the site I provided, the two reasons cited by this user completely sum up the situation as I see it - although it is obviously open to interpretation.