What is the source of the belief that the deceased become angels?

Within Christianity, the belief that humans become angels seems to stem from a misinterpretation of Jesus' words:

For in the resurrection they neither marry nor are given in marriage, but are like angels in heaven. (Matthew 22:30)
And Jesus said to them, “The sons of this age marry and are given in marriage, but those who are considered worthy to attain to that age and to the resurrection from the dead neither marry nor are given in marriage, for they cannot die anymore, because they are equal to angels and are sons of God, being sons of the resurrection. (Luke 20:34-36)

It seems many people incorrectly interpret "like" and "equal to" to mean "identical to."

Mainstream Christianity has never taught this; the early church fathers argued over many things concerning angels (whether they are only spirit or have material bodies, what categories of angels exist, etc.), but it doesn't look like they had to argue against the idea that men became angels. Not one of the thirteen commentaries I've read make the claim (or even leave open the possibility).

Outside mainstream Christianity, a few groups do teach that humans become angels:

According to the Latter Day Saints:

Therefore, when they are out of the world they neither marry nor are given in marriage; but are appointed angels in heaven, which angels are ministering servants, to minister for those who are worthy of a far more, and an exceeding, and an eternal weight of glory. For these angels did not abide my law; therefore, they cannot be enlarged, but remain separately and singly, without exaltation, in their saved condition, to all eternity; and from henceforth are not gods, but are angels of God forever and ever. (D&C 132:16-17)

And LDS teaching on angels:

There are two kinds of beings in heaven who are called angels: those who are spirits and those who have bodies of flesh and bone. Angels who are spirits have not yet obtained a body of flesh and bone, or they are spirits who have once had a mortal body and are awaiting resurrection. Angels who have bodies of flesh and bone have either been resurrected from the dead or translated.

The Swedish philosopher/scientist-turned revelator Emanuel Swedenborg, in his book Heaven and Hell published in 1758, wrote that angels are man's perfect form (they are the same race; angels are pure/purified humans).

... heaven is a communion -- sharing all it has with each one, and each one sharing all he has from this communion. An angel is a recipient, and is thus heaven in least form... Man, too, as far as he receives heaven, is so far a recipient, a heaven, and an angel. This is described in the Apocalypse in the words, He measured the wall of the holy Jerusalem a hundred and forty and four cubits, the measure of a man, that is of an angel (xxi, 17). [NKJV, ESV]

In Heaven: A History (1995: Yale University Press) the authors, Drs. Colleen McDannell and Bernhard Lang, state that Emanuel Swedenborg (1688-1772) had a pivotal role in bringing about a changed view of heaven, including the idea that angels are humans who have died and gone on to heaven, rather than a separate race of pre-created beings as mainstream Christianity had held for many centuries previously. See especially Chapter 7: "Swedenborg and the Emergence of a Modern Heaven."

Swedenborg based his teaching that all angels were once people primarily on his own claimed experience of meeting and talking with them in the spiritual world. For example, he says in Heaven and Hell #311:

People in the Christian world are totally unaware that heaven and hell come from the human race. They actually believe that angels were created in the beginning and constitute heaven, and that the devil or Satan was an angel of light who became rebellious and was cast out together with his faction, and that this gave rise to hell. Angels are utterly amazed that there can be this kind of belief in the Christian world, and even more so that people know absolutely nothing about heaven, even though this is a primary doctrine of the church. Knowing that this kind of ignorance is prevalent, they are profoundly delighted that it has now pleased the Lord to reveal to us so much about heaven--and about hell as well--and so as much as possible to dispel the darkness that is rising daily because this church is drawing to a close. So they want me to testify on their behalf that in all heaven there is not a single angel who was created as such in the beginning, nor is there in all hell a devil who was created as an angel of light and cast out. Rather, all the people in heaven and in hell are from the human race--in heaven the ones who have lived in heavenly love and faith, and in hell the ones who have lived in hellish love and faith.

Swedenborg also states, however, that this is the teaching of the Bible if the Bible is properly understood. For a brief summary, written from a Swedenborgian perspective, of the Biblical arguments in favor of angels and devils as human beings who have died and gone on to the spiritual world, see the article "What the Bible Says: Where Angels Come From."

Here are the basic points made in the article, accompanied by supporting citations from the Bible:

1. The Bible says that angels are "men" or "people."
2. Angels look like people.
3. Both are described by the same terms.
4. Angels themselves reject the idea that they are superior beings.
5. The Bible never mentions angels being created.
6. We become just like the angels after death.

For my own overview of and introduction to Swedenborg's teachings about angels (I am a Swedenborgian minister and scholar), including a section on where angels come from, see "Who Are the Angels and How Do They Live?"

Swedenborg's Heaven and Hell has been translated into many languages, and has had a large circulation throughout the world since it was first published in Latin, London, 1758. Outside of various statements in the Bible itself, it is the primary source of the common modern belief that people become angels after death.

However, Swedenborg did not say that angels have wings. Rather, he said that they are fully human in appearance.