Sometimes it's just the shorted form of the Eagle and Child which was the family crest of the Earls of Derby. They owned a lot of land in East London so there's a few pubs there called the Eagle and Child.

The Eagle was also the crest of Truman's Brewery (defunct) so many pubs formwerly owned by Truman's will be called The Eagle.

I'm sure there's many other reasons for it being used as a pub name.