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The 1/4 wave ground plane antenna seems to be the most common antenna for use on payloads. Its omnidirectional radiation pattern makes it suitable for use on a payload as it will emit equal radio power in the direction of all receivers.
A typical 1/4 wave payload antenna can be constructed from stripping 164mm of insulation for a length of 50 ohm coaxial cable (RG58). The copper braid can then be separated into 4 strands and secured in an X formation by either using tape to the bottom of your payload box, straws, or soldering solid core wire to the braid.
The core of your coaxial cable will act as the driven element of the antenna. If your cable came off a reel it will probably want to stay in a curved shape. Supporting the driven element of the antenna by placing it inside a drinking straw will prevent this, and should also give some support if the payload has a rough landing.
Example images of a 1/4 wave antenna (from http://www.stratodean.co.uk/search/label/Components): http://1.bp.blogspot.com/-mF2Kyo6-8GU/ULPJNIxeexI/AAAAAAAAA7I/IWLYiqOTld0/s1600/IMG_1638.JPG http://4.bp.blogspot.com/-R5zyNoIaHU4/UWrc2n6P6zI/AAAAAAAABTk/xsqZ0TyUdeU/s1600/IMG_2021.JPG
