What type of wi-fi antenna costs the least?
A general rule of thumb is that the higher an antenna's gain, the higher its price. Gain refers to the ratio of the output signal strength of an amplifier to the input signal strength and is usually expressed in terms of decibels (dB). 3 dB of gain doubles the power. This means that a transmitter outputting 200 milliwatts to an antenna having 3 dB gain produces 400 milliwatts effective power. Antenna manufacturers specify gain as dBi, referenced to a theoretical isotropic radiator. An isotropic antenna would form a perfect sphere, or radiation pattern, around itself, and does not exist in the real world. It is theoretical and is used as a base point for measuring actual antennas. It is a hypothetical antenna that radiates or receives equally in all directions. Isotropic antennas represent convenient reference antennas for expressing directional properties of physical antennas. For more information, get the ARRL Antenna Book: The Ultimate Reference, essential for anyone wanting to understand and construct antennas, without a lot of math.
Try making a commercial grade or an inexpensive 'downpipe' or a slotted waveguide. Look for a discarded Primestar (high gain, low cost, parabolic reflector with an offset feed) dish and build either the feed can or the BiQuad. Check out the New Zealand USB WiFi adaptor antennas, the Mali plastic water bottle antenna and the milk tub USB antenna. Take a look at a helix antenna. The wide angle sector antenna is a more challenging diy project. Yet another cantenna style is made with aluminum dryer vent pipe.
Be aware that there are additional factors of proper grounding techniques (earth ground to the antenna mast) and lightning protection devices (lightning arrestors) to protect wireless equipment, to consider when installing any antenna outdoors. Think about weatherizing the antenna with a radome. For a pringles or coffee can antenna the radome would the plastic lid that came with the can.
More accessible material is available at Paul Wade's W1GHZ Online Microwave Antenna Book. See mwave1 or mwave2 for additional information. Find out for yourself which design works best. Experiment with it!