In order to do this daily quest, you will need access to the Sevii Islands (by reaching trainer level 80 in Kanto) and have progressed the main quest up to the point where you have obtained the Rainbow Pass. Head to Six Island and proceed east to Water Path where you will then travel south into Ruin Valley. Speak to Archeologist Arthur at the bottom right of Ruin Valley to begin the quest.
After speaking to him, you will need to interact with seven broken pillars around Ruin Valley and battle the fossil Pokémon that are guarding them. After defeating all of the Pokémon, return to Archeologist Arthur. You will provide him with a word written in morse code that must be translated for him to understand. Decoding the word is very simple using the guide that he provides, which is shown below:
Pillar 1: Just above Archeologist Arthur
Pillar 2: Along the top-right side
Pillar 3: Along the lower-left side
Pillar 4: In the bottom left corner
Pillar 5: To the left of Morris
Pillar 6: Just below Ruin Maniac Stanly
Pillar 7: At the ledge leading back to Archeologist Arthur
Rewards: 25 Trainer EXP, $250 Pokédollars, Chance to receive an Aurora Ticket