what side of the neck the mane lays on...

I have honestly never heard of this... You will have to get some people on here that are more savvy with this kind of thing. But, my Arab has a super thick mane, like super thick. And, with all of its thickness is has absolutely no choice but for the thickest part to have to be on both sides of the neck. So I don't know about that.

I believe its supposed to lay to the right side. Not sure why, lol. Would like to find out though!

Well traditionally you are supposed to braid it on the right, but it doesnt naturally lay that way for many horses.

My horse has his on the left, I've tried to switch sides, but its to thick ):

I didn't think that there was a specific side that it was supposed to be on. I've heard some stories about the sides that manes lay on. One of them was that the side the mane lies on is the side that the horse favors the most. The other was that the leg on the side that the mane lays on is slightly shorter... Not sure how true any of those are, especially that last one. I actually don't think that there's any truth to the last one at all.

The first one, however may be true. My horse is left handed all the way. His mane lies on the left side... he prefers you to lead, and mount from the left side, he lunges better from the left side and more

I've only had anyone tell me about the mane during shows. Apparently it started in pleasure classes where there was only counter-clockwise riding, so the mane should lay to the right and not be distracting to the judge in the middle of the ring or so that they could see the horse's neck for whatever reason.
I can't say if that's true or not, because I truthfully don't care, but that's just what I've heard. :p
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I'd say it means about as much as the number of white feet telling you how reliable the horse is, the position of swirls on the neck telling you what his temperament is, the shape of the swirl on the forehead telling you how smart he is, and how far he rolls over telling you how much he'll bring at market.

It's all interesting bits of horse sense, with perhaps an element of correlation between physical attributes and mental/emotional tendencies, but take it with a large grain of salt. Just like people. I know some really laid-back redheads, some fantastically ditzy brunettes, and some scary smart blondes.

The one thing with the direction of the mane that I did hear that might be worth investigating is that, if the mane lays on one side for part of the length, and switches to lay on the other side for the rest of the length, that the neck could be out of alignment, "chiropractically." I forget where I heard that - not from a chiropractor, though. I can see that being possible, and if the neck is out of alignment that would definitely influence their posture and carriage.

Knowing Linda's "style," though, I'd expect it was more of a behavioral, emotional balance and carriage sort of thing, not a physical issue indicator. (Bad Scoutrider... I'll be nice now, pwomise... )

Scoutrider, that is a very good possiblity. Much better than the stuff I've heard, especially the one about leg length. Because in Psychology I've recently learned that extraordinary claims need extraordinary evidence. And the leg length is a very extraordinary claim. So far you have the best explanation regarding manes and the side they lay on or change to.