We eat Kugel on Shabbos based on the Fact that we got the Maan (RAMAH -Orach Chaim 242:1)
The question is why eat the thing in Memory of the Maan on the one day when we did not get any Maan? Also why is it not mentioned in Hilchos Yom Tov also?
The Mechilta in Parshas Beshalach says that on Shabbos the Maan tasted and looked different than the whole week. (What it means exactly - I do not know - as the Maan tasted based on what you wanted it to taste, so what was different on Shabbos? Also did the Shabbos portion change color on Shabbos or did it look different when you picked it on Friday?)
However, based on this Mechilta you can answer that we eat Kugel on Shabbos due to the fact that the Maan had a superior look and taste on Shabbos.
I saw another answer from the Toras Chaim the reason we eat all the foods we eat on Shabbos Kodesh is because it is similar to Moshiach coming which is Known as A day that is all Shabbos. Therfore we eat Meat as a symbolic gesture towards the Meat of the SHOR HABOR and Fish instead of the LEVYOSON we drink wine Instead of the WINE for the Tzaddikim. The Gemara in Chaggiga(YUD BEIS) says there are Seven Heavens one of them called SHECHAKIM that is it is the part of the sky where the Maan is prepared for the Tzaddikim in the future.Therfore it is this Maan that the Ramah is refering just as we eat the other food as a symbolic gesture for the Days of Mosiach so to the all important Kugel which represents the Manna which the Tzaddikim will eat in the times of Mosiach.