Low resonant frequency:
Drivers with a lower resonant frequency ( Fs less than 40 Hz) will make better subwoofer drivers in most instances.

Size Does Matter!
The lower the frequency, the more air the driver's cone has to move. Consider this - for every octave that you go down in the audio spectrum, your speaker has to move 4 times as much to provide the same output level! As a consequence of this, your subwoofer is probably going to end up being the largest speaker in your living room. 

Sturdy construction
Flimsy cones and driver baskets are likely to produce distortion, resulting in a less than perfect subwoofer design. To test the cone of a driver, push down one side of the cone near the surround. If the cone visibly distorts or doesn't move straight down into the basket, choose another driver.

Impedance
If you're planning to build a passive subwoofer (i.e. one that's driven by your main amplifier), then its impedance has to be the same or greater than the amplifier's minimum impedance rating.  If you plan to build a powered subwoofer, then impedance is less of a concern (most subwoofer amplifiers can easily handle 4 ohm impedances).

Please note that there is NO requirement for the impedance of your subwoofer to match the impedance of your main speakers!

Car Audio Drivers
Typically, car audio drivers will not do well as drivers for a home subwoofer system, as they are usually designed to produce a good response when a car's cabin gain is taken into consideration.  However, there are a few for which  the T/S parameters suggest that they can produce good results in the home.  IOW, if you're planning to use a car audio driver in your design, treat it as a home audio driver and design a system based on its T/S parameters to get the best results.

Using an existing driver
If you've already purchased a driver that you want to use in a subwoofer design, then chances are that the manufacturer has provided one or more enclosure designs for the driver. For the least amount of worries, it's usually safe to go with the manufacturer's recommendations. If a specification sheet was not included with the driver, try contacting the manufacturer directly. If the manufacturer has not provided recommendations for suitable enclosures for the driver, then you'll need to use the Thiele/Small parameters for the driver in order to determine the optimum enclosure for it. Sometimes these parameters are provided along with the driver, but if you can, you should measure them yourself in order to get the best results with your subwoofer design. .

Size Limitations
Do you want something small and unobtrusive or is something the size of a small refrigerator Ok? Smaller subwoofers tend to have higher cutoff frequencies, but the SAF is higher as well.  Subwoofer drivers typically require big boxes to perform at their best. 

Design Goals
Is the subwoofer going to be used as part of a small three-piece system that will be used mostly for popular music, or is it going to be part of a large home theater system? For the former example, a subwoofer that reaches below 45 Hz should be good enough, while the latter may require a subwoofer that extends below 30 Hz.

Room Gain
In most rooms, the room itself provides additional gain at frequencies below approximately 30 Hz. There's a slight risk of creating a boomy system if you create a design that on paper has a cutoff point much below 30 Hz. A commonly-recommended rule of thumb is to aim for a design with a response that is -3dB at 30 Hz and -9dB at 20 Hz, which should work well in most rooms.

Car Audio
Most cars provide an additional gain of approximately 12dB/octave below 80 Hz or so. Note that sealed and 4th order bandpass systems have response characteristics that naturally complement the natural bass boost provided by a car's interior. When designing a sealed subwoofer for car audio use, aim for a cutoff point between 40 - 50 Hz to achieve the best results.  For other types of systems, you will need to adjust the design parameters a bit to get the best results in-car.

Note: if you are using an existing driver, the driver's T/S parameters will typically make the choice for you. Low Q (<0.3) drivers are generally more suited to higher-order bandpass and horn-loaded systems (which are not covered here).   Drivers with Qts between 0.3 to 0.4 are usually best used in vented systems, and drivers with higher Qts are usually best suited for sealed systems. 4th order bandpass systems can usually work with drivers with any Qts between 0.3 to 0.6. In the case of car audio systems, you will almost always get the best results with a simple sealed system