Suffix ‘-ium’ vs. ‘-um’ in element names
I haven't found any common factor distinguishing the elements with names ending in -ium from the elements with names ending in another letter followed by -um. It seems fairly arbitrary, and in fact there was some variation historically between forms ending in -ium and -um for tantalum and lanthanum (and there is still variation between aluminium and aluminum in American English).
Most elements are metals. Many of the nonmetals are halogens or noble gases, which have different naming patterns using the suffixes -ine and -on respectively. But the nonmetals helium and selenium, and the metalloids tellurium and germanium have the -ium suffix also. The user J... left a comment saying that helium was originally thought to be a metal; a quote supporting this can be found in Linear Christmas's answer to the related Chemistry SE question Why do the names of most chemical elements end with -um or -ium?. The Oxford English Dictionary (OED) entry for selenium says that it and tellurium were "formerly classed among the metals".
Apparently, in Latin the names of metals always had the neuter grammatical gender ( Lectures on Syntax: With Special Reference to Greek, Latin, and Germanic, by Jacob Wackernagel, edited by David Langslow, published 2009). Examples of this are ferrum "iron," aurum "gold," cassiterum "tin" (a loanword from Greek κασσίτερος). Latin neuter nouns of the second declension regularly take the ending -um when singular and in the nominative or accusative case. (Janus Bahs Jacquet has pointed out in a comment that some metal names in Latin belonged to other declensions and so took other endings, such as aes "copper/bronze/brass." But I have the impression that new words formed in scientific Latin are generally in the first or second declension.)
So the -um part of -ium is just this Latin second-declension neuter ending. In Latin, -i- is a common derivational suffix (words formed with the suffix -i- end in -ia/-ius/-ium depending on their gender). Generally, Latin -i- is used to derive adjectives from other nouns (Wiktionary) but in Latin adjectives and nouns can often interchange, so it's often also possible to use words with this suffix as nouns. I haven't found a reference that explains why -i- shows up in so many of these element names; I'd assume it's because as new elements were discovered people named them in honor of existing things and the easiest way to derive a new name like this in Latin is to use -i-. For example, rhodium is apparently named after roses (the Ancient Greek word for "rose" is rhodon). Also, as more and more elements were named this way it became a noticeable pattern that people might follow even if they were not aware of its origins.
The OED doesn't provide an extensive explanation of the origins of the use of -ium, but it mentions the Latin metal names and gives some historical context for the English ones:
the names of sodium, potassium, and magnesium, derived from soda, potassa or potash, and magnesia, were given by Davy in 1807, with the derivative form -ium; and although some of the later metals have received names in -um, the general form is in -ium [...].
As I said, I can't see any factor connecting all of molybdenum, platinum, tantalum, and lanthanum. (The names tantalum and lanthanum were coined more recently than molydenum and lanthanum, so I have re-arranged your list to put them last.)
Molybdenum, according to Wiktionary, is derived from the Greek name for lead, μόλυβδος molybdos. The OED provides some additional details about how the -en- got in there: it references the earlier word "molybdena", from Latin molybdaena, from ancient Greek μολύβδαινα molybdaina "a kind of lead ore" which according to the OED ends in "-αινα, suffix forming nouns". The rule about metal names having neuter gender in Latin does not seem to apply to the names of ores: there are also words like galena (feminine) "lead-ore; dross", anthrax (masculine) "cinnabar", and (lapis) haematites (masculine) "red iron-ore/hematite" (it also had other meanings, apparently). I don't see any particular reason for why molybdenum was used as the name of the element instead of molybdenium.
Platinum seems to be from a Spanish word, platina, meaning "little silver." I guess it makes some sense to adapt this by simply changing the ending to -um rather than deriving a new word with -i-. But the process seems fairly arbitrary.
Tantalum (discovered and named in 1802) is named after Tantalus. I don't see any reason why it isn't tantalium. In fact, Wiktionary lists that as an obsolete spelling, and there is a question that was asked earlier on this site about that variant of the word: Reason why tantalium became obsolete.
Lanthanum (named sometime around 1840) is named after a Greek word meaning "to go unnoticed." I don't know why it doesn't have the ending -ium; in this case as well, it seems completely arbitrary to me. Wiktionary lists lantanium and lanthanium as obsolete alternative forms, and there is a question about them here: Lanthanum vs lanthanium.
I agree with semelic's answer for the most part, but that answer seems to overlook the obvious commonality for molybdenum, platinum, tantalum, and lanthanum -- and yes, aluminum (I'll come back to that) -- where the Latin and Greek root words lack the -i at the end of the stem. That qualification is not sufficient to explain everything, but the lack of terminal -i in the stem at least seemed to allow for a -um suffix in early element nomenclature.
(Note: Platinum was rooted in Spanish platina, as noted in the other answer, but that gave birth to a neo-Latin platina term for the metal in the 1700s, which Davy modified to platinum in the early 1800s to conform to the neuter singular second-declension endings for many other metals, as in ancient Latin aurum, argentum, ferrum, cuprum, stannum, and plumbum. The Spanish platina was ultimately rooted in the Latin plattus; none of these roots had any terminal -i. And the other -um elements all came directly from Latin or Greek roots without terminal -i.)
As mentioned in the Chemistry SE answer linked in the other answer, it was only in 1811 that the chemist Berzelius tried to establish that all new elements should be named with Latin forms. (And Berzelius seemed to prefer different language roots: hence we still live with symbols Na for sodium and K for potassium, because Berzelius tried to advocate for natrium and kalium, which came from then-recent Greek and Latin coinages from Arabic words, instead of Davy's roots of soda -- from Italian, based on Arabic -- and potash from Dutch.)
Thus, practice was only becoming standardized for element naming around the time many of these words came into existence. Molybdenum was named from a root with no -i in the late 1700s. Platinum, aluminum, and tantalum all were coined in the first couple decades of the 1800s from roots with no final -i. Other metallic elements isolated and named around this time often had a final -i in their classical root word. Or, in some cases, they derived from non-Latinate/Greek roots, in which case Davy and other chemists seem to have adopted the practice of using -ium by default around this time.
Note that this practice is not entirely consistent, as it wasn't until 2002 that IUPAC finally officially required the practice for new metal names to terminate with -ium. The practice in the early 1800s was not standardized yet, but it's pretty clear that the -um metals named around this time mostly derived from Latin/Latinate roots with no -i-.
And note that Latin was still a common written language among scientists at this time, so they'd be familiar with even less-common words in Latin. For example, as noted in the other answer, rhodium is indeed ultimately from Greek rhodon, so why the -i-? Well, any decent writer or speaker of Latin from that time would likely be familiar with derived Latin terms like rhodinus ("made from roses") or oleum rhodinum ("oil of roses"). It's therefore not a stretch to think they might throw in the -i- for a new word like rhodium. Not to mention that Rhodi- words were connected with the ancient Greek Rhodia and Rhodes, place names which historically were thought to be etymologically related to rhodon, the Greek for rose. (That etymology is likely not true, but it was historically thought to be connected, and puns on these words were common.) Rhodum, on the other hand, would literally be the Latin word for Rhodes in the accusative case, potentially creating confusion. If you're a chemist who knows and perhaps writes in Latin, rhodium -- implying something "of roses" (or "of Rhodes") -- is a safer bet.
You may not be able to explain away all the -ium endings before ca. 1815 this way, but many of them you can, along with the idea that -i- was commonly used when forming adjectives in Latin indicating the source of something (also probably related to forming the genitive plural of 3rd-declension "i-stem" nouns). Also, the potential for confusion with existing Latin words could have played a role, e.g., beryllium seemingly could be beryllum, based on Latin beryllus meaning "beryl," but beryllum would literally be the accusative of beryllus, potentially confusing the element with the source ore/gemstone beryl. Perhaps better to go with a pseudo-Latin derivation, following the third-declension genitive model to create beryllium, "of/from beryls." (Yes, this is all ad hoc justification here, so maybe it's not true. I'm just pointing out there are many good rationales to insert -i- for linguistic reasons.)
Which brings us to the aluminum/aluminium controversy. Again, as noted in the other answer, tantalum and lanthanum both developed popular historical variants with -ium. It seems with most recent new elements having an -ium suffix, there was a trend to try to apply this universally among some writers and speakers. (Platinum probably didn't suffer this fate, since as I noted above the element already had a very similar word in use, without an -i suffix. Molybdenum came about before the long trend of new -ium words started.)
As one more bit of evidence toward this, I note Davy's early attempt at naming the element derived from alum. At first, Davy tried alumium, appending the -ium suffix directly to the English word alum, since alum was the source. However, that term ran into objections from other chemists, so he proposed aluminum. Why? Like platinum based on recent Latin/chemical names, there was a recent Latin word alumina (1790s), in this case referring to the source ore. (Molybdenum was similarly derived from the source ore molybdena.) Alumina was based on the Latin root alumen rather than the English alum. When declined, alumen has the stem alumin- (with no terminal -i), so Davy went with the pattern of producing a neuter in -um, hence aluminum.
But an early reviewer of Davy's book noted that an -ium ending would have a "more classical" sound, whence aluminium, despite its somewhat more dubious etymology.
Clearly the practice was in flux at this time, but I believe the short answer for why quite a few early names for elements were chosen with -um is because they lacked the terminal -i in their roots (as did several extant Latin words for metals). But that -- perhaps inconsistent -- practice sensitive to original roots was rather short-lived (with popular -ium variants emerging in many cases, aluminium surviving as dominant in most of the world), and by the mid-1800s the modern practice of -ium for new metals had basically become standard.