The story
Compound of hand + egg. The jocular implication is that the name football ought to denote using the foot to propel a ball-shaped (i.e. spherical) object, which is the case for association football (also known as soccer, or “football proper” in this context), as opposed to using the hand to propel a spheroid (egg-shaped) object.
Etymology from Wiktionary, CC BY-SA 4.0.