Why did Frank let this character live?

I can only give possible interpretations:

• Frank was letting Harmonica live: as you point out, Frank only kills that child because he heard his name. His initial intention was to let him live. Thus he does not seem to automatically kill any witness to his criminal acts, especially if he does not consider him a potential threat. Similarly, it might be the case that he just decided to let Harmonica live because he was not his business, after all we never really know if or what was the reason for him to kill Harmonica’s brother. Given his character, he might also have wanted to keep Harmonica alive by pure sadism, to let him suffer the death of his brother.
• Frank wasn’t letting Harmonica live: again, as you rightly point out, during the flashback scene we see Harmonica falling to the ground due to exhaustion. They seem to be in the middle of the desert so it might be the case that, after Harmonica’s brother died hanged, Frank and his minions left, assuming Harmonica would also die there from the lack of food and water, or simply don’t caring at all about what would happen to him.

I don’t think the film ever clearly solves this point, and the viewer is left speculating.

Here is my minority opinion: we do not know if Frank allowed the boy to live, or if he killed him or, alternatively, left him behind to die. I always believed that Charles Bronson´s character Harmonica is not identical with that boy! We never truly learn Harmonica´s true identity. The boy in the flash back looks totally different and does not resemble Harmonica at all. Unlike Harmonica, who is clearly Caucasian, the boy seems to be Native American and/or Hispanic. Harmonica also seems to be too old for having been that boy! Sergio Leone paid so much attention to every small detail. Would he really be so careless when he choose the child actor for the boy? I think that Harmonica witnessed Frank´s callous treatment of the boy and his family - and he decided to hunt down Frank no matter what, and make him pay for all his killings! And it does not really matter who he is. We do know that the part of Harmonica was offered to other actors, before it went to Charles Bronson: Clint Eastwood was allegedly Leone´s first choice, but Eastwood didn´t want to be in yet another Spaghetti Western. And I think that is a good thing. Eastwood was not mature enough at the time. James Coburn was also asked, but he allegedly wanted too much money - which is a pity, because I love James Coburn! Anyway, all those actors are Caucasians. The boy from the flash back is not Caucasian!