Melon vines tend to sprawl all over the garden, taking up substantial space for a fairly long growing season. This makes melons ideal candidates for intercropping, which is a more accurate description of what many people call "companion planting." Intercropping or polyculture means planting two or more plants together, in the same bed or plot, in ways that benefit each other through association. Melon intercropping can follow traditional patterns or explore new ground.
Intercropping in Nigeria
Growing plantains, a tropical crop, requires relentless labor to keep weeds under control. Agriculture research at the University of Nigeria discovered that intercropping melon vines with plantains can completely eliminate the competitive pressure from weeds without setting back the plantains. Farmers could grow an extra, valued crop to help feed their families in the bargain. Another very traditional intercropping approach in northern Nigeria successfully uses melons for early weed control in larger plantings of yams, cowpeas and maize.
Intercropping in America
America has its own intercropping traditions, most notably "the three sisters," time-honored Native American crops that are grown together--corn, beans and pumpkins or other squash. In the three sisters system, the corn provides support for the beans, the beans add nitrogen to the soil for the corn, the corn serves as a screen against the vine borer, and the squash vines prevent weeds and shade the soil to conserve water.
Covering the Ground
An easy way to intercrop with melons is to generally follow traditional American and Nigerian strategies, growing melons to prevent weeds between rows or stands of another, taller crop. Corn is a good intercropping companion for the family garden, especially if you plant in rows and leave enough space to walk between rows. Go all out and add pole beans to the corn. Sunflowers could also work, though all parts of the sunflower, including roots, are somewhat alleopathic--they contain chemical compounds that discourage the growth of other plants--so make sure melon hills are well away from sunflower roots and that water doesn't drip from sunflower leaves onto the melons. You probably should keep all sunflower parts out of the compost pile, too.
For a totally different approach, provide trellis supports for your melon vines--encouraging vertical growth and saving garden space--and plant bush beans in between. There are other good possible associations, of course, but beans thrive in the summer heat, as do melons, fix nitrogen and are fairly unfussy and easily contained in limited spaces. Make sure that vines don't escape their supports and choke off your beans.