|La mano dall'inferno|
None of your drumming abilities are dictated by the hand you sign your name with or open doors with or throw a ball with.
In playing the drums there is usually a lead hand, which may or may not be the same hand you write with. That hand starts most things, plays the strong side of the rhythm, and generally gets the most practice. It also plays the cymbal rhythm, and coordinates most closely with the feet, so it can be a challenge to do that exact same things with the other hand.
That doesn't mean the other hand is “weak”, and using it is not a Sisyphean struggle against biology, as some purport. It simply is not as practiced. I have encountered exactly no players of any age with left hand problems that couldn't be addressed in a few weeks or months of the right kind of practice.
So everybody stop building failure into your language— and excusing your lack of practice— by calling it your weak hand. I have had it with that.
This isn't only our fault, or the fault of people marketing drumming systems based on you believing one of your hands is weak. This good hand/suck hand thing is baked into most languages from the beginning. Fairly benignly in English— the names right and left suggest the correct hand and the other hand or the left-over hand. German is similar, with rechts suggesting correct... and links somewhat ambiguous. It has the same ancestor as the English slink, but I don't know if it has that kind of slinking/scurrilous connotation to modern Germans.
Romance languages have the very old association of clean (or able) and dirty built into them, most plainly the Italian destra and sinistra— dextrous and sinister. In French the words seem to reference manners with droit and gauche, but the implied meaning is the same. Spanish has the screwball izquierda for left, which is borrowed from Basque, and I suspect it sounds as random to Spanish speakers as it does to us. Esperanto, which was supposed to be the language of universal peace and brotherhood has dekstra and maldekstra— basically, able and badly-able.
Other Indo-European languages mostly have the same working/dextrous/able and dirty/crappy/evil/weak thing going. One Old English thing I wish had survived was to use a euphemism for the left hand, and call it the friendly hand. It was embarrassing and indiscreet to speak openly of that dirty hand you clean yourself with, so they went the opposite way and called it the happy hand. That also happens in Greek.
I don't know the history of referring to one hand as “dominant.” A lot of Americans seem to be attracted to the word, and like thinking in those terms. I can't find any egalitarian names for the hands. I thought there might be an Asian language that gives them a Yin and Yang connotation, but there doesn't appear to be.
Drummers could call them the cymbal hand and the snare hand... a drumset-centric thing that would really irk those snare drum guys. The way the hands function practically in drumming, often we're dealing with a leading hand and an opposing or opposite hand. People who think we're supposed to aspire to perfect ambidexterity could call them hands A/B, 1/2, 0/1. If those are still too hierarchical, we could assign them any two random Greek letters. I suggest omicron (O) and chi (X).