The fairies put a mirror into Saphir's bedroom, and it showed not his own face, but a beautiful girl.
After he convinced it of his regret, it sent him back with a grain of gold, and told him to find a room and take off a maiden there, without heed to her resistance.
The frog sent him back with a grain of diamond and told him to find the garden, and cut off the branch with the bird he sought on it.
The names reflecting peridot and sapphire are purely for effect, but serpentine is unusual as it is dark, not a gemstone or particularly valued, and it is associated with serpents.
The connection may be to prime readers for the amphibian transformation, as to children both frogs and snakes are considered slimy, disgusting 'creepy crawlies'.
Going up in value, sand we all know to be worthless, and gold is not a gemstone, so the diamond will obviously work (also making it that three is magic, an old belief common to fairy tales).
The place of the mysterious bird, Constance, is unclear and seems to reflect another mythology or a related tale.