Of all the characters that DC reintroduced in the Silver Age, Aquaman has to be considered the most unique, for many reasons. First, he was not a "relaunch"; he was one of the few DC superheroes (along with Batman, Superman, Wonder Woman and Green Arrow) to have been published continuously from the Golden Age on.
Flash, Green Lantern, Atom and Hawkman had all been regular characters in the Golden Age whose series were cancelled and thus could be reintroduced to a new generation of readers with revised costumes, secret identities and origins. With Aquaman, DC didn't have that luxury, especially since he'd been a part of the Justice League of America from its introduction in Brave & Bold #28 (Feb-Mar 1960). Thus when they decided to give him a tryout for his own title with Showcase #30 (Jan-Feb 1961) they did not try a real relaunch; they just expanded his adventures.
He'd arguably been relaunched in Adventure #260 (May 1959), which created a new origin:
His mother turned out to be from Atlantis, and she passed on to her son the ability to survive underwater indefinitely, to mentally command the creatures of the sea, and to swim at amazing speeds. In Adventure #269 (Feb 1960), he gains a companion in Aqualad, an outcast orphan from Atlantis.
But this revealed another oddity about Aquaman compared to the other Silver Age characters. Aquaman didn't have a private life. He didn't maintain his Arthur Curry identity. He didn't have a home. He didn't have a girlfriend (unlike all the other Silver Age DC characters except notably Batman).
And so things stood when DC gave Aquaman his tryout for the big time with Showcase #30-33. The first story was blessed with the artwork of Ramona Fradon, one of the few female artists of the Silver Age, who had drawn Aquaman for several years in the short stories. The second story marked the beginning of Nick Cardy's historic run on the series.
Those looking for gay subtexts will not be shorted, although to me this almost seems to be implying a maternal role for Aquaman:
The early stories were all edited by Jack Schiff and suffer from his endless willingness to publish stories involving monsters at this time. No kidding, here are the villains in the first several stories:
The stories themselves are pretty mundane as you can imagine from those covers and they were book-length tales, another rarity for DC at the time. It's a struggle to find any continuing characterization efforts. Check out the "Aquacave" in this panel:
Those are pretty spartan digs.