You say "something isn't working right" but I doubt that anything is seriously wrong; it's probably literally just a normal phase he's going through.
Boys, before puberty, do not have a falsetto register as such. Boys before puberty sing alto or soprano naturally.
Falsetto is something that becomes possible after the onset of puberty, and after a boy's voice changes; after the hormonal development in his body changes his voice range into the lifelong adult range of either tenor, baritone or bass.
Learning to sing in falsetto, in terms of traditional, classical training, is something that usually happens after the age of 18 or so. In traditional, classical vocal study, the technique is called countertenor rather than falsetto. Most classical countertenors are in fact baritones, but some basses and tenors can learn this technique as well.
However, the falsetto or countertenor technique is also used in American R&B and soul and genres of popular music. I can't speak to that or how to learn it; I'm classically-trained. (I'm a tenor; I have not studied the countertenor technique.)
Update: If your son has merely "already started to go through puberty" then it's possible that his voice is still "cracking" or "breaking" and has not settled into a consistent adult voice type. The age at which this happens, and how long it takes things to settle down, can vary widely from individual to individual.
I'm also inferring that your son has never studied singing before, such as having sung in a choir before his voice began to change. So he has no basis of training to work with.
The only way to evaluate the situation is for your son to sing for a professional voice teacher, school choir director or music teacher, etc. and get some feedback on what his voice is going through. Depending on where he's at, the voice teacher may very well say that he should wait some number of months for his body to mature further before approaching learning to sing.
The human voice is a complex thing. You can't learn much of anything useful by reading written descriptions on a website. If you are really concerned about this, you need to have your son meet in person with a voice teacher.