Samuel went to the house of Jesse, as the Lord had commanded the judge and prophet to do, to anoint a new king for Israel. Samuel was impressed initially by the first son, but only by his appearance, and the Lord let Samuel that Lord had not chosen that one. Samuel would tell Jesse that the Lord had not selected that particular son, until Samuel had gone through seven sons,

Thus Jesse made seven of his sons pass before Samuel. And Samuel said to Jesse, “The LORD has not chosen these” (1Sa 16.10). 

Was there any symbolism in not finding a new king in the first seven sons of Jesse. I cannot see any, but one reader has suggested in the past, that eight symbolizes a new beginning, and David was the eighth son of Jesse, and David began a new and better reign as king after Saul died.

Therefore, seven appears to be incidental in this case, although eight might be significant.