Since The Changeling is the title of Middleton's play, the meaning and etymology of "changeling" plays an important role in understanding the purport of the drama.
Collins Dictionary defines "changeling" in its archaic form (as would have been prevalent in Middleton's era) as an idiot or as a person who is changeable in emotions or behavior, someone who is fickle. The mythological definition is a stupid, strange or evil child left behind by fairies or elves to replace a stolen charming, intelligent and good child. Both the mythological definition and the archaic definition have significant relation to the play.
The word is first recorded between 1545 to 1555 (of course it may have been in verbal usage before that). It is a derived word with "change" and "-ling" as the root and suffix forming it. "Change" is from the 1175-1225 Old French word changer from Latin cambīre meaning to exchange. "-ling" is a suffix that forms nouns and is from Middle and Old English -ling that was borrowed from German -ling. This is a pejorative or insulting suffix that indicates someone that is connected with something. In this case, a changeling is connected with one person being changed out for another person.
This word applies in two important ways to Middleton's play. One way is that Antonio fulfils the archaic meaning of idiot and acts the mythological part of a malicious elf or fairy and changes himself out for a fool in the asylum. He actually reverses changeling mythology by (1) being both the changeling and the changer and by (2) changing a stupid, worthless person for an intelligent, valuable person (himself).
The other way changeling is important is that Beatrice proves herself to be a changeable and fickle person and in a dangerous way. She, like Antonio, changes herself out for another persona as though she herself were the evil fairy or elf. She exchanges the kind, lovely, charming lady persona she appears to be the essence of for the persona of someone who can order villainy and do murder. As with Antonio, the changeling mythology is reversed by her actions: each wilfully changes their good self for an evil self instead of their good selves being stolen.
Alsemero, I am a stranger to your bed;
Your bed was coz'ned on the nuptial night,
For which your false bride died.
Forgive me, Alsemero, all forgive;
'Tis time to die when 'tis a shame to live.