In the original A New Hope, the rebel crest is only prominently featured on the helmets of the Alliance pilots. Something like the crest first appears in the design process in sketches made by John Mollo, the Academy-Award-winning costume designer for the early Star Wars films. Prior to working on the first film, Mollo had no specific experience in costume design, but he was an expert on military history and the history of military uniforms in particular. (And traditional French military uniforms, which Mollo had studied, featured the suggestively similar fleur-de-lis emblem.) After seeing the third draft of George Lucas's screenplay (at which point the story was still quite different from the one that eventually was used), Mollo started working on costume sketches.
Conceptual designer Ralph McQuarrie also made major contributions to the costume design, most notably the design of Darth Vader's masked and helmeted outfit.* However, I can find no evidence online of any A New Hope work by McQuarrie that featured anything like the crest on helmets or anywhere else, although he did experiment with helmet decorations in his concept work for snow troopers and Boba Fett for The Empire Strikes Back. I do not find any evidence of the crest in the sketchwork of visual effects art director Joe Johnston, who developed the vehicle and equipment designs for the film. (The entirety of The Star Wars Sketchbook featuring much of his work, is apparently available for download online.)
The first sketch of Mollo's that I can find that shows anything like the rebel crest shows a blobby red trefoil-like pattern on the helmets of two rebels.
Obviously, these are far from the final versions. The belt and holster designs on these two figures, for example, were ultimately used on Han Solo, rather than the uniformed rebels.
As the costumes developed, Mollo considered other kinds of crests, as shown here, for example. The helmet of the pilot on the left features a helmet with an entirely different design. (The figure on the right shows an interesting combination of the orange suit that was part of the final pilot costume for the movie, along with a helmet, jacket, and scarf that look more appropriate for a World War II flier, but it shows where the ribbed sleeves on Luke's snowspeeder pilot outfit came from.)
Thinking about Vader's design, Mollo also considered giving the Dark Lord (as well as some other imperials) a crest on his helmet or elsewhere on his costume. In these sketches, he tries out several different locations for the graphical elements, as well as several different appearances for the Sith crest, including ones with distinctly trefoil structures again. Stylistically, the emblem on the bottom right below bears a certain resemblance to the final rebel crest, although it still has a radial symmetry to its three parts, rather than tan the head-and-wings structure seem in the final product.
I have not been able to locate a costume sketch by Mollo that shows a version of the final rebel crest. However, based on stylistic evidence and the absence of any similar emblems in the artwork of the other major design artists working on the film, I think it is extremely likely that John Mollo originated the rebel crest design.
*This painting by McQuarrie, for example, may be the most famous piece of Star Wars concept art of all.