Suddenly it turned out that vacationing on a desert island wasn't the best idea. Also, why does she call her "mum"?
We don't know. Kyou clearly has a connection to Kiku from before Kiku woke up from cryo stasis. But the manga has left the details of that relationship an ongoing mystery.
Kiku is, as joked about repeatedly early on, technically 79 years old, and thus Kyou started out digging at her by calling her "Grandma" both because of her age and because she considered Kiku's general behavior to be trying to act like an older sister.
"Kiku" is the Japanese word for the flowers known in English as chrysanthemums, also called "mums" for short. After bonding and agreeing to work with her, Kyou started calling her "Mum," short for "chrysanthemum" and also a generation lower than "Grandma" (and thus closer to "older sister.")
Kyou explains this when she gives the nickname on Chapter 4 Page 20.