Quarter vs. half-wave
A quarter-wave antenna, in the sense of "a wire or rod 1/4 wavelength long", is not a complete antenna; it is half of one. The other half is whatever is connected to the other terminal of your transmitter output — probably its ground, i.e. the ground plane or chassis (if any) of your board. This works well as long as there is a quarter-wave-or-longer surface adjacent to the antenna feed point (the place where the two connections spatially diverge from each other, also the place you start measuring the length from).
A half-wave antenna is a good choice when your entire device is much smaller than a quarter-wavelength, or when you are building an antenna that is physically separated from the transmitter.
Given that you mentioned placing components inside of the coil (this may not be actually a good idea depending on the specifics) I presume your device is small and therefore a half-wave antenna would be wanted.
Coiling an antenna element — I assume you mean a helix, not a flat coil (loop antenna), which has yet different properties — is one way of shortening it. A shortened antenna is always less efficient than a full-length one.
With a small transmitter (board/chassis ≤ 1/4 wave), if you have a choice between a helical half-wave and a straight quarter-wave occupying the same space, you want the half-wave because otherwise the ground for your quarter-wave is too small so you have an even worse shortened antenna.
With a large transmitter, a quarter-wave antenna is a better choice because the overall space taken up is less.
(Note that if the device has any cables of any sort running to it, they may also act as part of the antenna system unless you specifically design so they won't. This can be an advantage in saving space or it can be an interference problem, depending.)