In Pāḷi texts, the right shoulder is not specified. The standard description of a monk meeting the Buddha says
ekaṃsaṃ uttarāsaṅgaṃ karoti (Vin 1.46)
he arranged his robe over one shoulder.
However the right side of the body is generally emphasised, e.g.
Atha kho so, bhikkhave, mahābrahmā ekaṃsaṃ uttarāsaṅgaṃ karitvā dakkhiṇaṃ jāṇumaṇḍalaṃ pathaviyaṃ nihantvā... (DN ii.36; cf. MN i.168, i.177 etc.)
Then indeed, monks, the Great God, having arranged his robe over one shoulder, placing his right (dakkhiṇaṃ) knee-cap on the earth...
Atha kho rājā māgadho ajātasattu vedehiputto ... uṭṭhāyāsanā bhagavantaṃ abhivādetvā padakkhiṇaṃ katvā pakkāmi.
Then indeed Ajātasattu Son of Vedehi, King of Māgadha... rose from his seat, saluted the Bhagavan, and keeping to the right (padakkhiṇaṃ katvā), he left.
The verb many of these is √kṛ - "doing, making". So ekaṃsaṃ uttarāsaṅgaṃ karoti literally means "he makes his upper-robe one-shouldered" and padakkhiṇaṃ katvā means something like "making right-handed".
The connotation of auspiciousness with the right side of the body is something shared with the Romans from whom we get words like "dextrous" from dexter "right, right hand side, south (facing the the rising sun)"; and it's opposite "sinister" (literally "left). In fact the Pāḷi word dakkhina is cognate with Latin "dexterous" (they both come from the same Proto-Indo-European root). And just like it, it can mean "skilful, clever, and quick to learn", but also "lucky, auspicious, etc".
One explanation might be deduced from the modern day Indian practice of using the left hand to wash the backside after defecating. If this is an ancient practice, which it may well be, then the left hand was ritually (and factually) polluted and the right hand ritual pure.
However, I cannot find a single reference to the robe going over the right shoulder in either the Nikāyas or the Vinaya. It is always simply ekaṃsa "one shoulder". So this is apparently a later tradition.
NOTE Apparently Stack Exchange doesn't allow me to post Chinese characters! So I'm having to give Romanji instead which is much less satisfactory!
I tried to check the Lapis Lazuli citation, but he doesn't give enough information to locate the particular sūtra in the massive “Dàbǎojī jīng” or Mahāratnakuṭa - it runs to 120 fascicles in the CBETA version of the Chinese Tripiṭaka. But plenty of other Mahāyāna texts use this kind of phrase.
Ěr shí ānán cóng zuò ér qǐ, piān tǎn yòu jiān, yòu xī zhuó dì, (T232 729.b17)
Then Ānanda rose from his seat, bared his right shoulder (yòu jiān), and knelt on his right knee.
Now we can search for the phrase "yòu jiān" and see when it started appearing in Chinese translations. It turns up in some early translations, e.g. T156 which is an Avadāna Sūtra translated during the Wei Dynasty (220-265 CE); in T1428 and T1432 some of the Chinese Vinaya Texts. It also turns up in the Āgama texts. So this suggests that the introduction of specifying the right shoulder occurred after the closing of the Pāḷi Canon (which occurred quite early), but still fairly early.