The Mermaid’s Pendant is a special item obtained from the Old Mariner for 5,000g after achieving 10 hearts with a marriage candidate villager, giving that candidate a bouquet, and having the first farmhouse upgrade. The Mermaid’s Pendant is used to marry a villager. The Old Mariner is on the beach past the bridge (can be repaired with 300 wood) and will only appear on rainy days. A Rain Totem is the only way to summon him during Winter.
The Old Mariner will appear while you are married but will not sell you another Mermaid’s Pendant unless you get divorced.