If there was anything specific to the J-visa program, I would expect to find it in the CFR Title 22 → Chapter I → Subchapter G → Part 62. Searching there for "30 days" does not reveal that much helpful stuff, though.

@pnuts At University of Washington, the Out of Country Request form for absences longer than 30 days (available at ap.washington.edu/cms/wp-content/uploads/…) includes a certification that "* J-2 dependents will depart the U.S. with the J-1 Exchange Visitor." The condition to complete this form is that "the Exchange Visitor anticipates being outside the U.S. more than 30 days". I guess that rules out the OP's second option, unless he does not anticipate to be out of the country for that long a time.

@pnuts Having faced the possibility of a more-than-30-days absence myself, I have learned that you need to file a "notice of absence" with your sponsor. Here's such a form: global.upenn.edu/uploads/media_items/… Combining this requirement with my interpretation of "should the J-1 principal be out of the US for more than 30 days at a time (...) [which the J-1 is from the first day of his absence, in my opinion], the J-2 dependents cannot remain in the US (...)", I concluded the J-2 has to leave with the J-1.

Probably, the J-2 has to leave as soon as the J-1 lets the sponsor know his intentions to return only after more than 30 days. It might still be a path to make option 2 work. The question is whether doing so will jeopardize the return of J-1 and/or J-2 into the program.

It seems clear to me that the OP's first option works if a couple of weeks is not longer than 30 days. Now, the second option will not work if he tells his sponsor that he will leave for four months - the J-2 will have to leave with him. I was wondering what will happen if the OP plans to leave for not longer than 30 days (as in option 1 - the J-2 can stay), but his stay turns out to be longer than 30 days (as in option 2 - the J-2 would have had to leave in the past) while he is away.

Nice find. This does not really answer whether the second of the OP's options is possible, if it is known that his stay will be longer than 30 days before he leaves. It sounds as if it is not possible (that is, the J-2 has to leave with the J-1 when the J-1 leaves for a known absence of more than 30 days).

@Berwyn another example is having "PHD" appended to your last name (by the United booking website, in this case). This is more difficult as it impacts your ability to check in (and so forth) with booking code and last name. But United said it won't be a problem flying.

Google Flights can do this, too. And very easily so, in fact, so easily it does not deserve it's own answer :)