Origin of "come in handy"

It hails from the 1800s (per Dictionary.com) and it means:

Be useful or convenient, as in This check will really come in handy.
[Mid-1800s] Also see come in, def. 4.

It can be explicated come in to the situation and be handy, where handy derives from hand, in the sense of right-hand man - useful.

Here is the Ngram graph of its usage:

This led me to an 1840 usage of the expression:

A useful implement close at hand is how I would understand something handy. Most times we have to draw inference from precedents and usage rather than hard and fast definitions.

Early use of the phrase seems to indicate such an intention on the part of the authors, as well.

Sometimes, though, come in handy may have been used in one or the other of the meanings alone: useful or accessible.

It looks like it evolved from the Middle English hende, which carried both the meaning of "readily accessible" and "useful" as well as a host of other definitions, which itself rose out of the Old English/High German gehende, which, interestingly enough has more or less the same definition as "handy" does today. I just wrote a paper on it, but I wasn't able to find any confirmation, so this may as well just be treated as food for thought or one of many possibilities. As it happens, hende was used in Middle English literature as of the late 13th/early 14th century, so the evolution into "handy" could have happened at any point between its own recorded uses and then.

Unlike the other answers, I'd suggest that this idiom is a composite.

Handy needs no explanation in its sense of "useful" or "convenient".

To come in is being used idiomatically in the sense of "become" or "eventually be", perhaps by connection to other senses in which come in is used to mean "arrive" (like come in last in a race, come in dead to a morgue, etc.)

When declaring that something will come in handy, it usually means that the thing is recognised to be useful in general, but perhaps not yet for a specific purpose (or not for a purpose immediately occurring), so it will be useful only later.

Often one could say that something might come in handy, to emphasise uncertainty about the later application.

When one refers retrospectively to something that came in handy, it might be to express that it's usefulness or convenience wasn't obvious ahead of time.

In his answer Daniel traces the early records of the whole phrase “to come in handy”.

It is a simple collocation of “to come in” and “handy”

[All quotes from the OED]

Although it is not the earliest recorded written use, one assumes that the actual the earliest use was

Handy 2. a. Made or done with [or by] the hands; involving manual labour or skill. Now: spec. designating minor domestic repairs or maintenance, such as might be carried out by a handyman. Cf. handiwork n., handicraft n.

1535 Bible (Coverdale) Hag. i. B Vpon men and vpon catell, yee and vpon all handy laboure. [Upon men and cattle, and indeed upon all manual labour] (my translation)

And “associated with the hands”:

1586 W. Warner Albions Eng. ii. vii. 26 Then settle they to handy Armes. [handy armes = hand-held weapons]

The earliest use is only approximately 10 years earlier:

1. a. Clever with the hands, dexterous; skilled at a variety of (manual) tasks; able and ready †to do something or turn one's hand to anything. Cf. handyman n.

a1525 Bk. Chess l. 1034 in W. A. Craigie Asloan MS (1923) I Ane duke thare was, ane worthy handy man, That daylye faucht and..wan. [There was one duke, an honest and skilful man, that fought and won on a daily basis.”] (my translation.)

By the early 17th century, “hand” had already developed a use to indicate the convenience of having that skill and the readiness.

3. Useful, helpful, convenient.

1616 R. Bernard Dauids Musick To Rdr. sig. A3v To propose what we haue..in that method and forme, which may be best & most handy to the readers vse.

This then takes us to to come in as a phrasal verb:

16. intransitive. To prove to have a specified positive quality.

a. With adverb or prepositional phrase, as usefully, well, etc.

1733 T. Stackhouse New Hist. Bible I. vi. iv. 909/1 His Father's losing his Eye-sight, by the hot Dung of Swallows, had been a sad Family-accident, had not the Gall of the Fish come in opportunely to remedy it.

(The OED's earliest use of “to come in handy/useful" as a whole phrase in the OED is a plagiarism/reprint 30 years later of that given by Daniel above.)