The word Aiko has many meanings, but the most popular and the one applied to Chapter 624 is likely....
愛子 — Love and child, child of love
It basically means "love child".
Now what does Naruto Chapter 624 have to do with that?
The chapter starts off with Hashirama and Madara splitting because of the feuds between their clan, with Madara talking about how their dream was nothing but idealistic trash. Then it ends with Madara giving the ultimatum of Hashirama killing himself or killing his brother to realize this dream.
What is this "dream"?
Their dream is to create a village where there would be peace and children wouldn't die.
Aiko is referring to the village as being their "love child" (lol) between Hashirama and Madara, referring to the way that they were the founders of the village, it was their ideals that would make Konoha.
tl;dr Aiko is a "love child" and Konoha was the aiko of Hashirama/Madara.
愛子 — Love and child, child of love
It basically means "love child".
Now what does Naruto Chapter 624 have to do with that?
The chapter starts off with Hashirama and Madara splitting because of the feuds between their clan, with Madara talking about how their dream was nothing but idealistic trash. Then it ends with Madara giving the ultimatum of Hashirama killing himself or killing his brother to realize this dream.
What is this "dream"?
Their dream is to create a village where there would be peace and children wouldn't die.
Aiko is referring to the village as being their "love child" (lol) between Hashirama and Madara, referring to the way that they were the founders of the village, it was their ideals that would make Konoha.
tl;dr Aiko is a "love child" and Konoha was the aiko of Hashirama/Madara.