The next element in the sequence is:
Why? Because each item in this sequence corresponds to:
The order of Periodic Table elements in the song 'The Elements' by Tom Lehrer (1959), in which Lehrer sings the names of 102 elements of the Periodic Table to the tune of "I Am the Very Model of a Modern Major-General" from Gilbert and Sullivan's 1879 comic opera The Pirates of Penzance. (Bolded text in the title of this puzzle, note...)
By considering the elements in Periodic Table order, the nth term of the sequence corresponds to the ordered position in the song lyrics of the nth chemical element, like so:
5 (5th position in the song) = Hydrogen (element 1 in the Periodic Table)
51 = Helium (2)
47 = Lithium (3)
48 = Beryllium (4)
37 = Boron (5)
97 = Carbon (6)
7 = Nitrogen (7)
6 = Oxygen (8)
56 = Fluorine (9)
91 = Neon (10)
The next member of the sequence should therefore correspond to the position of the 11th element of the Periodic Table - Sodium - in the song. Sodium is the last one mentioned - in position 102.
Of course, at the latest count there are actually 118 elements in the Periodic Table, which is why many may consider the sequence needs an additional 16 elements, as per Hint 1...
If you're interested in calculating further terms of the sequence...
I recommend you watch the linked video! But also here are the lyrics to Lehrer's song, with the positions of each term of the sequence relevant to this question emphasised in bold, and every tenth element numbered for ease of counting:
There's antimony, arsenic, aluminum, selenium,
And hydrogen and oxygen and nitrogen and rhenium,
And nickel, neodymium, neptunium, germanium,
And iron, americium, ruthenium, uranium,
Europium, zirconium, lutetium, vanadium,
And lanthanum and osmium and astatine and radium,
And gold and protactinium and indium and gallium,
And iodine and thorium and thulium and thallium.
There's yttrium, ytterbium, actinium, rubidium,
And boron, gadolinium, niobium, iridium,
And strontium and silicon and silver and samarium,
And bismuth, bromine, lithium, beryllium, and barium.
There's holmium and helium and hafnium and erbium,
And phosphorus and francium and fluorine and terbium,
And manganese and mercury, molybdenum, magnesium,
Dysprosium and scandium and cerium and caesium.
And lead, praseodymium and platinum, plutonium,
Palladium, promethium, potassium, polonium,
And tantalum, technetium, titanium, tellurium,
And cadmium and calcium and chromium and curium.
There's sulfur, californium and fermium, berkelium,
And also mendelevium, einsteinium, nobelium,
And argon, krypton, neon, radon, xenon, zinc and rhodium,
And chlorine, carbon, cobalt, copper, tungsten, tin and sodium.
These are the only ones of which the news has come to Hahvard,
And there may be many others but they haven't been discahvered!