We say every morning "Neshalma parim sefatenu". Since we cannot offer Korbanot , we got the Tfila instead. So the Tfila follows the order of the Korbanot. For the Korbanot the day starts after sunrise.
In the introduction to the Tur (upon which the Shulchan Aruch is based), R' Yaakov Baal HaTurim writes that he organized the Sefer based on the order of the day, how a person should conduct himself from when he wakes up until he goes to sleep.
Avraham initiated Shacharis, Yitzchok initiated Mincha, Yaakov initiated Maariv. The order of the Siddurim is in the order they were initiated. In addition Shacharis is considered the main Tefila of the day. Also practically Mincha / Maariv are often said immediately one after the other.