This is a community-wiki answer where we can compile a list of names and their meanings.

Tai-kun, Japanese. This name has a double meaning:

"Tai-kun," as written, is applying the "-kun" suffix for "boy" or "young man" to the name or nickname "Tai." This is consistent with the book's description:

Like Sniper, Ando used the Japanese nickname 'Tai-kun,' an old one, remnant from when the Child first appeared in the media's eye, riding wide-eyed on Hotaka Ando Mitsubishi's shoulders through the eternal overload of Tokyo.

However, the word 'taikun' translates as "liege lord; shogunate." So even his name of boyish endearment is also one of power and lordship.

Jagmohan, Indian. Jag is the world, and Mohan is to captivate, delight, enchant, etc - combined, the name means "one who captivates/enchants the world." It's a title of Lord Krishna, who is one of the most important avatars of Lord Vishnu, the protector in the Hindu Trinity. The name itself is not an unusual one in India.

(Is there any further significance to this one? This all fits well thematically. But in the chapter "The Tenth Director," "Jagmohan" is referred to as "practically a pun," which I don't understand.)

Xiao Hei Wang, Chinese. In Mandarin, xiao means little, and is often used as a term of endearment, usually for someone younger, and is added to the beginning of someone's name. Hei means red, a color associated with good luck and fortune in Chinese culture. Wang is a common Chinese surname, and can be translated to king. It would appear as if "Little Red King" is the Chinese strat's preferred nickname.

Micromegas, Utopian, sometimes abbreviated to "Mike."

His Utopian nickname is not short for Michael, though the invocation of Heinlein's might be intentional. It is short for Micromegas, "Littlebig," the alien visitor from Jupiter who towers over humankind in grandeur and philosophy in Patriarch Voltaire's famous (and possibly Earth's oldest) science-fiction tale.

(The Michael/Heinlein reference refers to Heinlein's alien visitor Valentine Michael Smith, of Stranger in a Strange Land.)

The Porphyrogene, Masonic. Meaning is something like "child of the purple," with the purple being the imperial color, signifying royalty.

The significance of the name "J.E.D.D. Mason" itself is a crucial plot point in Too Like The Lightning. "J.E.D.D." stands for Jehovah Epicurus Donatien D'Arouet - sacrilegious in our own day; even worse in the Seven Hives, where public mention of religion is forbidden.

Madame D'Arouet explains that each of J.E.D.D. Mason's ba'pas picked out one of the names.