Mamoru Hosoda was inspired to make this film when he asked someone about raising children. The person said raising children was like "having a monster or a wild animal in the house.
Ame and Yuki stand for "rain" and "snow," respectively, in Japanese. Both realized what they wanted to be in both weather scenes (Ame when they were running in the snow and fell in the river, and Yuki when she was stuck in school because of the rain).
Ame is the Japanese word for Rain.
Yuki is the Japanese word for Snow.
Hana is the Japanese word for Flower.
Yuki means snow, and she was born when it was snowing. Ame means rain, and he was born when it was raining.
In one scene, while Yuki is in class, the teacher talks about the flying speed of an unladen swallow - a reference to Monty Python and the Holy Grail.