Mazal tov! Your case appears straight-forward, although we request a list of the things he does with each hand and those he does equally with both. Then we can give a final ruling. First, allow us to present a little background.
All agree that a righty lays tefillin on his left hand, but three different prooftexts are cited as the source of this determination.‎{&&1}‎ The Tanna Kamma says the word “yadcha” (“your hand” – referring to where the tefillin are to be fastened) means the left arm. R. Natan derives it from the hekeish (juxtaposition of p’sukim) that associates the hand one uses to fasten the tefillin with the hand one uses to write a mezuza. Most people write with their right hands and, thus, use their right hands to fasten the tefillin onto their left arms. Yet, a lefty would use his left hand to fasten the tefillin on the right hand. Rav Ashi learns from the extra letter “ה” in “ידכה” (“your hand”)‎{&&2}‎ that tefillin go on the יד כהה (weaker hand), usually the left. A lefty’s right hand is the weaker one.
A major machloket exists among the Rishonim about one who writes with one hand and does most other activities with the other. The Sefer HaTeruma says to lay the tefillin on the overall weaker hand, without special emphasis on writing. Rav Yechiel of Paris says that one who writes with his right hand lays tefillin on his left arm, even if he does everything else with his left.‎{&&3}‎ The Shulchan Aruch‎{&&4}‎ brings both opinions, but he and the Rama favor Rav Yechiel’s opinion that writing is the determining factor. Thus, your case should be simple – your son puts the tefillin on his right arm.
However, despite the stature of the Shulchan Aruch and Rama, some major poskim question their p’sak. It appears that the two opinions in the Rishonim are based on two of the three sources in the aforementioned gemara, one of which stresses writing and the other general strength/skill. The Gra‎{&&5}‎ demonstrates that the majority of opinions follow Rav Ashi that we place the tefillin on the overall weaker hand. Furthermore, the Bach‎{&&6}‎ disagrees with the Shulchan Aruch’s understanding of R. Yechiel’s statement. The Bach says that R. Yechiel accepted both the source of “writing-fastening” and that of “the weak hand,” and one lays tefillin on the right hand only if he is a lefty in both regards. With a twist on this approach, R. Moshe Feinstein‎{&&7}‎ understands that one who writes with one hand but does most work with the other is deemed ambidextrous (sholet b’shtei yadav), who lays on the left arm.‎{&&8}‎ The exact parameters of sholet b’shtei yadav are not fully clear and, according to certain opinions and in certain cases, may cause one who considers himself a lefty to be treated like a righty. For example, there are different opinions about one who writes with both hands but prefers his left or one who writes script with one hand and print with the other. Therefore, we ask for more detailed information and hope that the situation will turn out clear-cut; otherwise, there may be no easy way to cover all bases.