This is a good question because it comes with a lot of good answers and suggestions, conventionally, all the main answers will involve suggesting some sort of combination of working on scales and arpeggios, octaves, trills, etc and some exercise book(s) from Hanon or Czerny or Bach or what have you.
I also believe that a good exercise is a mental exercise of being keen to the musical score (what is written on it). Understanding all the nuances of what the composer wrote (phrasing, rhythms, articulations, dynamics and pedaling notations, etc) and trust me, for pieces like , let say, a typical Beethoven sonata movement, you will have so many musical ideas involved even in the stretch of a couple of measures.
It is of course vital to learn the notes of the score, and to eventually get it up to speed, but I guess what excuses, for most students, the disregarding working on the musical details of the score , is because one wants to get a piece up to a certain speed and have it pass for a performance or recital, which is always in a few short weeks. To spend time really trying to work out all the nuances of the musicality written out in the score will definitely lengthen time spent overall on the piece, which doesnt bode well for most young students.
At least two musical technicalites in a score: dynamics and phrase markings... if spent some real laser focus on getting those right , will do wonders for one's actual expression of that piece. Dynamics and correct phrasing (and thus pedaling to accomodate for hashing out the phrasing) is the difference between real magical music that turns heads and keeps the attention of listeners/captivates the listeners , and a series of notes played out that is merely reminiscient of music, and unfortunately will make most students in a classroom fall back to looking at their phones during the performance.
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One idea on how to gain that musical keenness is by really knowing how your piece sounds like. Play it everyday that is performed by top performers of the piece ( and a variety of performers, so you get a picture of the musical range on how to express it) and hum/sing it to yourself (whether out loud or in your head). Do it a lot. Singing and humming the piece will allow you to kind of have an organic feel for the actual phrasing and dynamics of the piece (rather than "oh, i have to remember there is a pp in the beginning of this phrase" or "remind myself there is a staccato on these two notes in measure 45). You will want to really copy that sound in your head into your playing..thus you pay attention to the score more.
Topic: What are the most important exercises for piano students? (Read 2834 times)