Which is correct: "home in" or "hone in"?

I've heard people say "Home in on something", but I've also heard others say "Hone in on something".

Which is the correct expression, and what is the etymology of these?

Home in is correct. It bears resemblance to the concept of "home" in that the projectile (or a figurative counterpart) involved is getting to where it is meant to go. However, the variation "hone in" has increasingly been accepted to mean the same thing.

Also see definition #21 here of home, and this very helpful study on the two commonly confused expressions.

This reports the following:

Traditionally, a missile homes in (not hones in) on a target. Hone means "to sharpen." The verb home means "to move toward a goal" or "to be guided to a target." But some usage guides (see notes below) now recognize hone in on as an acceptable alternative to home in on.

An Ngram illustrates the wider usage of "home in on" than "hone in on", at least in current literature.

@zenbike ... I was a Soldier and I never heard "hone in on target" while I was on active duty.

From the OED on my computer and online:

verb [ no obj. ]
1. (of an animal) return by instinct to its territory after leaving it: a dozen geese homing to their summer nesting grounds.
• (of a pigeon bred for long-distance racing) fly back to or arrive at its loft after being released at a distant point.
2. (home in on) move or be aimed toward (a target or destination) with great accuracy: more than 100 missiles were launched, homing in on radar emissions.
• focus attention on: a teaching style that homes in on what is of central importance for each student.

From etymology online:

home (v.) 1765, "to go home", from home (n.). Meaning "be guided to a destination by radio signals, etc. (of missiles, aircraft, etc.) is from 1920; it had been used earlier in reference to pigeons (1862). Related: Homed; homing. O.E. had hamian "to establish in a home".

You can hone your skills but you home in on a target.