IBM components are given 3 or 4 digit identifiers as part of the module name and msg ids used.
IEFx is used for the initiator; IEF is used for device/space allocation services (don't know if there are others that fall under IEF).
IEFBR14 was the name given to a 1 (later to become 2) instruction pgm that does nothing but return control to the CALLer; hence the BR14. The 2nd instruction was added when users found that unpredictable condition codes were returned after execution. The reason: register 15 wasn't set to zero before return. The result: whatever was in reg 15 when IEFBR14 was CALLed was used as the return code. I've heard there were other changes, but don't recall them.
IEFBR14's purpose was to enable users to allocate/deallocate files by invoking the OS step initiator/terminator as any executed pgm would. Once they got it working properly, it served its purpose well.
You can use any pgm to do this. Just add an extra DD card to the step. But using IEFBR14 is less confusing.
The "IEF" derives from a convention on mainframe computers that programs supplied by IBM were grouped together by function or creator and that each group shared a three-letter prefix. In OS/360, the first letter was almost always "I", and the programs produced by the job management group (including IEFBR14) all used the prefix "IEF". Other common prefixes included "IEB" for data-set utility programs, "IEH" for system utility programs, and "IEW" for program-linkage and loading.
I guess I told you more than you wanted to know.