A piece of the action
A share in an activity, or in its profits.
'A piece of the action' has an unambiguously American flavour. It brings to mind images of gangster movies with Jimmy Cagney and the like demanding 'hey, gimme a pieca da action'. When the Star Trek franchise opted for a mobster themed episode in 1968 they called it 'A Piece of the Action'. It isn't essentially a US phrase though and tracing its genesis takes us well outside the USA and into a history of finance.
In the early 1600s, the Dutch came upon an interesting trading innovation - the company. Until then, the spice trade had been profitable but small scale, with spices being brought back from 'the Indies' (broadly what we now call Asia) along the tortuous Spice Road on pack horses. The high price of spices encouraged entrepreneurs to build ships to bring the spices back in larger quantities. There was big money to be made, but the large capital cost of building a fleet and the threat of loss from pirates made it too risky a venture for an individual investor; so in 1602 they formed a company - the Dutch East India Company.
Dutch citizens were invited to invest in the company, which had been given exclusive trading rights to half the world and tax-free status back home. Profits were huge and the clamour to invest was intense. Dirck Bas Jacobsz, one of the company's founders, was instrumental in managing the joint ownership by offering what were then called in Dutch 'acties' or, in English, 'actions'. These were certificates that promised a share in any future profits of the company and what, not unnaturally, came later to be called share certificates. These shares were often purchased by groups rather than individuals. What each of these good citizens had bought was literally 'a piece of the action'.
The term 'action' continued to be used in that context well into the 19th century and is still so used in Spanish (acciónes) and French (action). It was first recorded in English in John Evelyn's Diary, published between 1641 and 1706:
"African Actions fell to £30, and the India to £80."
'A piece of the action' is certainly a 20th century American phrase. Despite its 1930s mobster overtones, the first use of it that I can find is in the 1957 film Monkey on My Back:
"You want a piece of my action, Sam?"
The 'action' in the phrase means 'a share in an activity; an opportunity'. It is doubtful that whoever coined it in 1950s America knew the history of the Dutch East Indies Company but, knowingly or not, it was the Dutch 'acties' that were the source of that meaning of 'action'.