For this complicated puzzle, you'll need the note you got from the fountain in the ruins, the one labeled "Charlotte's Sketch" in your inventory. Read to the end of this hint for more help.
The locket has ten buttons arranged in a circle; the sketch shows ten letters arranged in a circle. It's not a giant leap of the intellect to see the connection here.
The bigger problem is figuring out what to do with those letters. The sketch says, "Mom, 1867, Right," but you can't spell any of those words with the letters you have. So maybe those words are just clues?
Look at the family tree to find that Charlotte's mom was Marianna -- and that name you CAN spell out with the letters you have.
And when you do use the sketch to spell out Marianna, two of the center buttons pop down. Aha, so we're on the right track!
You can find the answers to the other two clues in the family tree as well. Do you see 1867 anywhere on the family tree? It only appears in one place --
-- it's the birthdate of Hiriam Thornton. So enter Hiriam (notice, it's not Hiram, but HIRIAM) next.
Finally, what about "right"? Is it someone on the right side of the family tree?
Well, yes -- but one of those people on the right is more right than the others.
See Emeline Wright? That's too close to be a coincidence.
So the three names you need are MARIANNA, HIRIAM, EMELINE.
This is a forgiving puzzle; if you get a letter wrong, just start that name over again. Once you finish a name correctly and the center buttons go down, they stay down, even if you back away from the puzzle.
Still having trouble? Here's a picture of the locket with the letters marked; that might make it a little easier.