The naming of the elements

Traditionally the discoverer named the element, and they named it after place, mythology, some property of it, or whatever took their fancy. Since 1947 the world body called IUPAC, the International Union of Pure and Applied Chemistry, has been the arbiter of names. Discoverers now have the right to submit a name for approval, but it is IUPAC who decide.1

In 1949 they adjudicated on several rival names. So beryllium was chosen in preference to glucinum or glucinium, and niobium in preference to columbium. They adopted the spellings lutetium and promethium instead of lutecium and prometheum.

They also chose wolfram over tungsten. However, this did not last, and official usage has varied. In their official list wolfram is now treated as an acceptable variant, listing the element as tungsten (wolfram).

In 1990 they finally tackled the more contentious spellings, and declared that aluminium, caesium, and sulfur were the official names. But in the first two cases the alternative is still valid: they list them as aluminium, aluminum and caesium, cesium. However, they do not list sulphur, I don't know why.

Most of the element names are much the same in all languages, but ancient ones like gold and tin are obviously quite different in each language. So for uniformity these ancient elements are also officially known by their classical names. They are listed as:

    gold (aurum)
    silver (argentum)
    copper (cuprum)
    tin (stannum)
    lead (plumbum)
    mercury (hydrargyrum)
    antimony (stibium)
    potassium (kalium)
    sodium (natrium)
The last three were added in the 1990 review, and the last two aren't genuinely classical, but were coined back around 1800. These (and wolfram/tungsten) explain all the cases where the symbol doesn't fit the name.

Note that there is no general push to rename the -um elements as -ium, although all future elements will end in -ium (except the last two groups in -ine and -on. Molybdenum, platinum, and tantalum are safe. Aluminium was originally named alumium in 1807, then quickly became aluminum and finally aluminium, which was the form used by the American Chemical Society until 1924.

The transuranic controversy

The new transuranic elements went smoothly up to element 103. In fact the IUPAC rejected the Swedish claim to have discovered nobelium, but accepted their name for it anyway. Then in the 1960s the heavier elements were so fleeting that claims of discovery were dubious, and for several decades the IUPAC declined to adjudicate. So rival names sprang up.

The Soviets were first to claim discovery of elements 104 and then 105, which they called

    104   kurchatovium   Ku
    105   nielsbohrium   Ns
Later the Americans claimed to have discovered them, and called them
    104   rutherfordium  Rf
    105   hahnium        Ha
There was a lull, and once more discoveries were reported, they weren't given names. They were referred to simply as Element 104, Element 106 and so on. After a while the IUPAC devised a system of temporary names in 1990. New elements were to be designated using the syllables un, bi, tri, quad, pent, hex, sept, oct, enn, nil. By mixing Greek and Latin they got unique initials, which could be used to give the symbol. So Element 104 became unnilquadium, symbol Unq. (A double -ii- would be dropped, so 112 would be ununbium, not ununbiium.) These are only provisional names, and don't affect the right of the discoverer to propose a final name.

Eventually they got around to assigning permanent names.2 They recognized the American claim to having discovered

    106   seaborgium     Sg
and the German claims to
    107   nielsbohrium   Ns
    108   hassium        Hs
    109   meitnerium     Mt
but objected to the name seaborgium because Glenn Seaborg was still alive and to nielsbohrium because there was no precedent for combining first and last names. They also chose different names where there were existing rival claims. So in 1994 the IUPAC chose these names:
    104   dubnium        Db
    105   joliotium      Jl
    106   rutherfordium  Rf
    107   bohrium        Bh
    108   hahnium        Hn
    109   meitnerium     Mt
But there was much complaint and horse-trading about this, so they changed their minds, and in 1997 decided on these names, which are now universally accepted:
    104   rutherfordium  Rf
    105   dubnium        Db
    106   seaborgium     Sg
    107   bohrium        Bh
    108   hassium        Hs
    109   meitnerium     Mt
A perhaps unfortunate consequence of the IUPAC's decision is that the rejected names cannot be reused for subsequent elements, so Kurchatov, Hahn, and Joliot miss out on this particular immortality. For this reason the IUPAC ask claimant discoverers not to use the laboratory's own choice of names in the literature, but to stick with systematic provisional names like ununoctium or Element 108 until the right of naming is established. They also don't want a previous symbol for a rejected name to be used, so when copernicium was proposed (see below), the original proposed symbol Cp was eventually rejected as it had once been used for a claimed element cassiopeium.

The newest transuranics

Everyone has been cooperating since then, which means it takes a long time between the claim of discovery and the final proposal and acceptance of a name. Between 2003 and 2016 the remainder of the period has been confirmed as discovered, and officially named.
    110   darmstadtium   Ds
    111   roentgenium    Rg
    112   copernicium    Cn
    113   nihonium       Nh
    114   flerovium      Fl
    115   moscovium      Mc
    116   livermorium    Lv
    117   tennessine     Ts
    118   oganesson      Og
Note that the last two of these reverse an earlier decision that all future names will end in -ium. In the end, they decided that the one under astatine should be tennessine, and under radon should be oganesson, to match the name of the rest of those groups (except helium).

The IUPAC allow four possible sources of names: a place, a scientist, a mythological concept, or a quality of the element (such as for radium; none of the more recent elements have used this source).