I'm trying to understand the best share and how it relates to luck. What qualifies as a good best share?
A "best share" is nothing more than the largest share your miners have contributed. It doesn't have any actual correlation to anything (other than "bragging rights"). A "good" best share is relative to hashrate and time. At the current difficulty, a
truly good best share is one that's greater than 1,196,792,694,098.
It's as CG says here, it is only for bragging rights, but if you wish to know what a good best share is then the following explanation will help you understand better.
After 100 shares, on average your best share should be 100.
After 1,000,000 shares, on average your best share should be 1,000,000.
After 1,200,000,000,000 shares, on average your best share should be that too, meaning you should have found one block with it. That's why on average it takes current diff number of shares to find one block.
If your best share is less than the total number of shares you've contributed, then your luck is a little low, and vice versa. It has absolutely zero bearing on your chance of finding an ultra lucky best share in the very next share, but it's interesting to watch nonetheless.