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 a length of insulation from some 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 length of exposed core (driven element) and braid (radial) is calculated by the speed of light / frequency. This will give you your wavelength (as c=fλ). Dividing this number by 4 will give you 1/4 of your wavelength as required for a 1/4 wave antenna. However it should be noted that the velocity of a wave is slightly slower through an antenna than it would be in free space. So for this reason you should multiply your calculated length by 0.95

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 S_Mark http://www.stratodean.co.uk/search/label/Components):

Some good information about antenna design and propagation can be found at http://www.idc-online.com/technical_references/pdfs/electronic_engineering/Antennas_and_Propagation.pdf

You will need :

Wire cutters, sharp blade, ruler, Single core wire, RG174 coax. Example uses RG174 to SMA pig tail from HAB Supplies., Solder and Soldering iron :