How did a "ton" come to mean "one hundred" of something?

Speaking from my reasonably intact memory of the 1960s, we reserved the word ton for 100mph (outside of its proper meaning as 20cwt or 112lbs). If Fred had coaxed his Norton motor cycle to travel at 100mph along our first motorway (freeway), you might have heard some awestruck witness exclaim, "Blimey! Last night, Fred did a ton up the M1 on his Norton!" but otherwise, the British usage of the word, to mean £100 or in other non-standard contexts, was pretty much restricted to London and was rarely encountered in the rest of our green and pleasant land.

The sources below, including The Patridge Dictionary of Slang, cite as first slang usage of ton meaning one hundred the amount of £100 from the first decades of the 20th century. From there, ton was applied to different contexts with the meaning of 100 (miles per hour, cricket etc.). The original usage of ton referring to £100 appears to be from Cockney Rhyming Slang, a word construction which relied both on phonetic but also on semantic links between words “in which case the person coining the slang term sees a semantic link, sometimes jocular, between the Cockney expression and its referent”.

£100 at that time were certainly “a large amount (tons)” of money and this possible semantic link with the “100 cubic feet” usage might have found its way into common speech.

From World Wide Words:

A ton is £100 (half a ton being therefore £50). This relates to a common usage in a number of contexts, for example, to do a ton is to achieve a speed of 100 miles per hour and in darts or cricket a ton is a score of 100. This is familiar enough not to seem an odd usage, even though ton is most commonly met with as a largish unit of weight.

Actually, all down its history it has been a measure of volume as well as weight, perhaps not surprisingly so because the word comes from tun, the name of a type of wine cask, which could be treated as either. For example, the registered capacity of a ship is measured in volume units, not weight, in which a ton is taken to be 100 cubic feet (this is probably the origin of ton meaning 100, but nobody seems to know for sure).

Also the Green’s Dictionary of Slang appears to support this assumption:

ton n.1

[SE ton, 100 cubic feet]

1. [late 18C+] a very large (unspecified) amount; thus tons n.
2. [late 19C; 1940s+] £100.
3. [1950s+] 100 miles per hour; usu. as do a ton v., to drive at that speed.
4. [1960s+] any unit of 100, e.g. 100 years, 100 runs (in cricket).

This is slightly speculative in nature, trying to 'join the dots' from known facts to create a coherent narrative. Interestingly, it does seem that the Register ton might be the origin of using 'ton' to mean 100.

The etymology specifies the origin of the word to not just be from "cask" but from a specific measure/size of the cask.

From Etymology Online: "measure of weight," late 14c. The quantity necessary to fill a tun or cask of wine, thus identical to tun (q.v.). The spelling difference became firmly established 18c. Ton of bricks in the colloquial figurative sense of what you come down on someone like is from 1884.

This measure of wine has historically been roughly equivalent to 1000 liters in volume and around 1000 kgs in weight. As a unit of mass, the ton has always been used to refer to things in the 1000X range.

Very importantly, tuns (the cask) were also used as a measure of taxation of cargo ships. The modern word 'tonnage' is a measure of the cargo-carrying capacity of a ship and is used to assess fees on commercial shipping; historically it was the tax on 'tuns' of wine. Eventually, the decision was made that rather than assessing the tax based on the number of tuns the ship carried, it would be based on the overall capacity of the ship. This measuring of ship capacity was defined in 1854 in units of 'register tons' which was equivalent to 100 cu ft. 2

This is the only 'official' use of 'ton' as a measure which equates it to a 100 (of something). Collins Dictionary dates the use of the word 'ton' to mean 100s from the 20th century itself, so the 1936 usage of that meaning, does seem to be the earliest. Given that sources agree that this usage stems from cockney slang and that the cockney slang dates from after the definition of a register ton as 100 cu ft, it is possible to make the following leap in logic:

That the people who worked on the docks with ships, specifically the loading and unloading of cargo ships would be most familiar with the use of 'ton' to mean '100 (cu ft)' and hence the adoption of the word into slang would reflect this association and make it equivalent to '100 (of something)' rather than '1000' with which it is associated almost everywhere else.