Samuel appointed a time for Saul, the new king of Israel, that he should wait for Samuel before making an offering,

“You shall go down before me to Gilgal; and surely I will come down to you to offer burnt offerings and make sacrifices of peace offerings. Seven days you shall wait, till I come to you and show you what you should do”  (1 Samuel 10.8). 

However, when the battle was about to begin, and Samuel had not shown up, Saul started to lose control,

Then he waited seven days, according to the time set by Samuel. But Samuel did not come to Gilgal; and the people were scattered from him (1 Samuel 13.8). 

Why did Samuel set a time of seven days? The record does not say. Therefore, we cannot attach any significance to it in this case, according to my opinion.