Use a comma if the information is not necessary to the sentence. Leave the commas out if the information is necessary for coherency.
He lived in France with his wife, June.
Here, it is not essential to know the name of his wife–it’s just a bit of extra information.
He lived in France with his wife June.
This guy had three wives. The reader knows their names and needs to know that this refers to June, not one of the other wives.
It is up to you the writer to decide whether or not the names of his family are nonrestrictive (omittable) or restrictive (essential). I would consider the names of close relatives to be vital to a short biography; therefore, I’d skip the commas. This Chicago Manual FAQ page seems to agree with us. Scroll down to the question about “my wife Lucy.”
If you do go with commas, be sure to use semicolons for clarity as ultrafilter recommends.
He now lives in Lakeville with his wife Teresa, their son Zach, and daughters Leah and Persephone.
This may help solve the problem. Combining the two daughters’ names makes them the only possible daughters (I think), so the commas are unnecessary all around, as all names are now closely linked to the word they rename.
Clarity of style triumphs over rules of usage at all times. The plethora of commas “required by the rule” in the example makes it harder, not easier, to read. Technically “his wife, Eleanor” would be correct in isolation unless it was necessary to distinguish between “his (present) wife Eleanor” and “his first wife Marguerite.” But in the construction given, the excess commas do not add clarity but rather make the sentence more difficult to read, and should be omitted.
Also remember the apocryphal dedication to “my parents, God and Ayn Rand” – which is always given as a horrible example when discussions of the serial comma come up – as exemplifying why use of the comma requires common sense rather than adherence to style manual rules.
That said, it’s often wise to avoid appositives when this can be done, and in particular to exercise great caution in the “preceding appositive” where a title is used more or less as a modifier of the proper noun following it. Time Magazine is notorious for abuse of this principle: “Former Democratic and Farmer-Labor Party Governor of Minnesota Harold Stassen…”
Get rid of the word "their " for a more parallel structure: He now lives in Lakeville with his wife Teresa, son Zach, and daughters Leah and Persephone.
Or: He now lives in Lakeville with his wife Teresa and their children, Zach, Leah, and Persephone.
Some might phrase that: He now lives in Lakeville with his wife Teresa and their children: Zach, Leah, and Persephone.
But this changes the meaning of the sentence a bit.
With this wording, it is unclear it Teresa is the mother of Zach & the daughters, or if they are the husbands’ children from a prior marriage. Your other rewordings are better in that regard.
That is true. But it is awkward and unparallel to have “his wife Teresa” and “their son Zach” followed by a simple “daughters Leah and Persephone.” One of these things is not like the other. And “with his wife Teresa, their son Zach, and their daughters Leah and Persephone,” while parallel, is still awkward because it’s so freakin wordy.
But this rewording loses the info telling readers which of the children are sons or daughters. (Being in Minnesota, we could probably guess from the names for Zach & Lean, but Persephone…?)