The song was originally the ending theme for an anime titled "Miyuki"（みゆき）which was adapted from the manga series by Mitsuru Adachi（あだち充）. If anyone seeing the anime has noticed that the character design looks strangely familiar, it's because Adachi is even more famous for the baseball comic "Touch" that also became popular the year after "Miyuki".
Since then, it has become a favorite at graduation ceremonies since the lyrics talk about a young girl's eventual transformation into a woman and the memories that she leaves behind her. In fact, at a site called www.rankingbook.com, there is a list of the most popular graduation songs with H2O's magnum opus coming in at 31. No. 2 is "Okuru Kotoba" (贈る言葉)by Kaientai (海援隊) (which I showed the day before) while No. 1 is "Sotsugyou Shashin"(卒業写真) by Yumi Matsutoya(松任谷由実) (nee Arai). When "Omoide" was released in March 1983, though, it had gotten as high as 6th place on the Oricon weeklies and ended up being the 20th most popular song of the year.
The lyrics were by Yoko Aki (阿木耀子), who had written several of Momoe Yamaguchi's (山口百恵) hits back in the 1970s. And the music was by Kisaburo Suzuki (鈴木キサブロ）, who had written Mariko Takahashi's (高橋真梨子) "for you"; it's a very sentimental ballad, kinda reminiscent of AOR band America in delivery, guaranteed to bring out the water works from the lassies...and probably a few lads as well.
As for the video here, I found a tiny sliver of the ending credits for "Miyuki" at 5:06 with the shorter version of "Omoide".