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I have googled this one and seen arguments for both sides.
Say your piece would imply that you've had the opportunity to make your part of the statement on the subject.
Say your peace would imply that you've had the opportunity to speak and set your mind at ease.
Clearly, they both make perfect sense in those contexts. To be clear, I want to know which is the form used by most people?
It seems to be say your piece and hold your peace. Because 'peace' and 'piece' are homophones, they are easily confused.
Google Ngram shows piece being 20X more common than peace in 2008, but 45X more common in 1999 (acknowledging the horrendous predictive value of Ngrams). So, in a decade or so... it may well be; not based on the Ngram, but based on how misuse has so often turned into common usage.
I have always seen "say your piece," as if telling an actor to recite their script.
Other than that, I found that "speak your peace" can be used, but only in a very limited context, where "peace" literally means "peace." The general form uses "piece." This article also mentions that "speak your peace" is usually used as a variant of "hold your peace."
I am only familiar with 'speak your piece' in the sense of 'I gave him a piece of my mind', or 'a piece of advice'.
I have never encountered the 'peace' spelling outside a church context, when in the ritual of 'pax vobiscum' people exchange a sign of peace. Perhaps the 'peace' example here could be a mutilated reference to this, otherwise I fail to see how you can say you 'peace of mind', although you can definitely seek, and attain it, as well as 'get something off your chest to have some peace of mind'.
Furthermore, looking at the source link in the answer above, it's full of inconsistencies and errors - 'peace' is confused with 'piece' in the first answer slide, which doesn't help matters at all.
I interpret "piece" as "your thoughts on the subject" or perhaps, just "your thoughts". Except in a religious or quasi-religious context I can't conjour up any instance where "say your peace" would really make much sense. Perhaps after "saying your piece" you would attain peace of mind, but that's as close as I can get.
