If your son otherwise qualifies to be your dependent according to the U.S. tax rules (see this from the IRS), then your living abroad and your husband claiming him in the Japanese system likely isn't relevant (unless he’s a little older, in which case the question of who’s supporting him becomes important). Basically, if he’s also a U.S. citizen (or resident), is under 19, and lived with you without earning significant income, he’s probably your “Qualifying Child” (which is the best version of a dependent, for tax purposes).
There are possible child-related tax credits (some of which are affected by other tax variables related to your expat status). And with a dependent there’s also the possibility of “Head of Household” filing status (with lower tax rates), but that depends on who provided support. The TurboTax program will ask the questions which are usually needed to clarify this, but since an expat scenario with a non-resident spouse doesn’t always fit neatly into the rules, you may want to post here if you’re not sure about anything.
Speaking of your non-resident spouse, you could also elect to treat him as a U.S. resident if you wanted to (as these rules tell us — the IRS link doesn’t work very well, and you have to scroll up). There would be a few headaches involved, such as getting an “ITIN” number for him, filing by mail with an election statement the first time, and reporting all of his worldwide income on your joint U.S. tax return (possibly the biggest headache!).
The ultimate question would be whether the joint return with its additional tax benefits would outweigh the reporting of the additional income. Sometimes creating hypothetical tax returns to test it both ways is a good idea.
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