Learn English – A word that means “to become one’s name”

OED has this for eponym:

1.a. One who gives, or is supposed to give, his name to a people, place, or institution; e.g. among the Greeks, the heroes who were looked upon as ancestors or founders of tribes or cities.

and this for protonym:

The first person or thing of a certain name; something from which another person or thing takes its name.

The distinction appears to be that with eponym, the later thing takes its name from a direct relationship with the earlier thing; whereas with protonym the second thing has simply been given the same name as the first.

Wikipedia gives an example:

A synonym of eponym is namegiver (not to be confused with namesake.) Someone who (or something that) is referred to with the adjective eponymous is the eponym of something. An example is: "Léon Theremin, the eponymous inventor of the theremin."

OED has this:

1880 Scribner's Monthly Mar. 667/2: "The wrecked canal-boat, the Evening Star,..quenched in the twilight, with its heavenly protonym palpitating in the vapor above it."

The instrument called the theremin takes its name from its eponymous ("name-giving") inventor; whereas the boat Evening Star was simply given the existing name of an unrelated thing.

In the question, David Beckham is the protonym of the child.

In British English, the word namesake is a "bi-directional" relation: David Beckham and the child are namesakes. It appears from the Wikipedia entry that in American English namesake can mean protonym as well, and imply a uni-directional relationship.

How about moniker for the noun?

And I'd be inclined to invent monikerize for the verb.

(I also like appelation, but I don't have the right to up-vote it yet.)