It's really tough because if you're going to consider moon phase as a legit factor, then you also can't ignore things like day of the week, season, or ever what direction you're facing. (Although there may have been people who feel that any or all of those are proven/disproven.) That's another thing - a lot of people
believe different things. On top of that, usually when people care about it, it's on "important" or expensive synths - really hard to get anything close to a good sample size.
I've been trying to record these kinds of effects while cooking. There's no way I was going to cook thousands of something that's doesn't make money. The thing is, even though cooking is much cheaper per synth than say, smithing, and I move product pretty damn fast... in 2414 synths, I've on managed to craft on 5/8* named moon phases. Once that's separated by day, it's even more bleak.
*Even less when I narrow it down by skill level.
So the following are my results so far, divided up by the moon phases I've covered. In all of these, I am level 104, and using:
Craftmaster's Ring
Artificer's Ring
Hocho
Culinarian's Apron
Culinarian's Hat
Synthesis Support
Stall
and making pizzas. Here's what I know:
So, so far, waxing gibbous is giving me the fewest breaks and the most HQs. But, if I separate that moon phase out by day, I get:
Saying that Waxing Gibbous on Lightsday = HQ, and Waxing Gibbous on Darksday = no breaks.
BUT... look at those sample sizes! Not even close. Heck, haven't even given Iceday a chance - not to mention the other four named moon phases. I figure I'd want about 1500 of each in order to make any kind of useful guess.
All I'm trying to say is that a slight change one way or another can depend on a lot of factors, that may not even really be factors at all: phase, day, direction, element of crystal, whether or not Bastok is in last place and you're standing on one foot... That's why I'm doing this on pizzas... it's kinda fun to play with, and I cook pizza anyway.
Also: I don't want to take away from what people prefer doing - or make think something I come up with may be better. I'm just playing around with it because it helps keep crafting interesting for me.