Why is Batman just Batman while Robin is the Boy Wonder?
The name "Robin, the Boy Wonder" was partly based on the medieval hero of the poor, Robin Hood. The name also fit the flying animal motif of Batman.
In 1940, Robin was introduced as part of an effort by DC Comics to soften the dark broodiness of the Batman character. In time, virtually all of DC's major superheroes wound up with a "kid sidekick" of some sort. Aquaman had Aqualad, Green Arrow had Speedy and Superman had Jimmy Olsen as a vehicle for better plot development.
In the case of Robin, the phrase "Boy Wonder" was simply used as a marketing tool to emphasize his youthful contrast to Batman. The partnering of Batman with Robin the Boy Wonder has been one of the most successful match-ups in comic book history.