My understanding is that the "Murder, Inc" moniker was invented by the media, and the actual organization never used that name.
Based on reports in declassified FBI files and contemporary reports in the US media, it seems likely that the members of "Murder, Inc." referred to themselves as "the Combination", although we will probably never be able to be entirely certain about this.
I'm not sure that it is possible at this distance to say with absolute certainty what the members of "Murder, Inc." called themselves - if, indeed, they had any internal name at all. It may very well be that the sources simply do not exist.
However, given that caveat, we can certainly say something about how "Murder, Inc." was known to different groups, in particular to journalists and to members of US law-enforcement, and what those groups believed the members of "Murder, Inc." called themselves. We also have some reports from witnesses and informants who stated that the group was known to them as "the Combination".
It seems clear from information in the file about Abe Reles, released by the FBI under the United States Freedom of Information Act, that law enforcement agencies had already adopted the term "Murder, Inc." to refer to the group as early as June 1940.
A few months later, in September 1940, Life magazine published an article about "Murder Inc.". In it they state that the syndicate was:
.. known among its founders as "The Combination"
Perhaps unsurprisingly, under the circumstances, they did not cite their sources for this particular fact!
However, in a 1958 monograph about the history and activities (and, indeed the existence) of the Mafia, the FBI also said that:
"... witnesses revealed the existence of a vast criminal syndicate which the underworld referred to as the "Combination," and to which the press gave the appellation "Murder, Inc."
- [Section II, p27]
On a related note, the monograph also mentions an early version of "Murder, Inc." that had been uncovered by police in 1921:
"... in the form of a Mafia gang which had been given the name "Good Killers". This group of assassins was suspected of some 125 unsolved murders of Italians in New York City, Pittsburgh, Detroit, and Chicago."
- [Section II, p48]
So it seems that the FBI and contemporary journalists believed that the members of "Murder, Inc." called themselves "the Combination". We can also say that, based on the FBI file, this was the name that witnesses and informants provided to law enforcement agencies at that time.