The song he's trying to parody consists almost exclusively of the actual names of the elements. There are a few filler words added to make the meter work out ("There's strontium and silicon and silver and samarium"), and a couplet in English at the end, but otherwise it's just the elements.

The rhyme is trivial, since every line ends with "ium". To do it with anything else you'd need to work a lot harder.

As noted by a sibling comment, he's not rhyming the -iums, he's rhyming the antepenultimate syllable (selenium/rhenium, germanium/uranium, vanadium/radium, gallium/thallium, rubidium/iridium, samarium/barium). But the other thing he leans heavily into is alliteration. The third verse is chock full of this:

Holmium, helium, hafnium, erbium (all h, save the rhyme)

Phosphorous, francium, fluorine, terbium (all f, save rhyme)

Manganese, mercury, molybdenum, magnesium (all m)

Dysprosium, scandium, cerium, cesium (all s sounds, save the first)

Lead, praseodymium, platinum, plutonium (most p sounds)

Palladium, promethium, potassium, polonium (continue p sounds)

Tantalum, technetium, titanium, tellurium (t sounds now)

Cadmium, calcium, chromium, curium (k sounds now)

There's also the point to make that most element names are 3-4 syllables long, so the verses are structured with 4 elements per line. The extra-long praseodymium and neodymium are paired with the very short lead and nickel respectively, for example. In fact, prior to the last verse, the 4 elements per line is violated just once (last line before the intermezzo crams in "bismuth, bromine, lithium, beryllium, barium"). The last verse goes a regular 4 elements, then 3, and then launches into this:

* argon, krypton, neon, radon, xenon, zinc, rhodium

* chlorine, carbon, cobalt, copper, tungsten, tin, sodium

The first of these lines rhymes all the -ons at the end, and the second goes for alliteration instead.

Of course, there are a few failures to align all the elements by alliteration or rhyme. The ones in the last verse align instead by other means (first you have californium, fermium, berkelium--all named after the same place; then you have mendelevium, einsteinium, nobelium--all named after scientists). Now that's clever!

>"Lead, praseodymium, platinum, plutonium (most p sounds)" The mildly subtle humor herein since "Lead" has the symbol "Pb", as it comes from the Latin 'plumbum'. Thank you, Mr Lehrer.

That is an excellent close reading. Very astute. Thank you.

TIL a new word: antepenultimate. Quite lovely.

... and now Google has introduced me to preantepenultimate and propreantepenultimate. Wow.

I'll admit that I had to look up what the proper term was for before-penultimate, since I've never had reason to use it before.

It's not quite that trivial, since rhymes are based on the last stressed syllable in a word (e.g. he rhymes "samarium" with "barium", but not "lithium").

But isn't the elements song itself a parody of "Modern Major-General" which has plenty of non-rhyming words (although of course with a lot of rhyming words too)?

Sure. It's just that if what you want is to do a Pokemon-based version of this song, your ideal would be to cram it as full as possible with Pokemon names. You could step back and do a Pokemon-based G&S parody, rather than a parody-of-a-parody, but it doesn't have quite the same panache.