A good Honeydew Melon is a real treat. Bright mint green, with juicy, crisp honeysweet flesh. For best flavor, choice Honeydews that weigh 5 lbs. or more. Look for a waxy white to cream rind that is barely tinged with green. When fully ripened, the Honeydew exudes a slight honey aroma, and the blossom end should give to a little pressure.

Honeydews should be well-shaped and  nearly spherical and uniform in appearance. There should be no scars or surface defects, and no evidence of bruising. They should appear heavy for size, the surface waxy and not fuzzy. 
U.S. grades are No. 1, Commercial and No. 2. Distinction among grades is based predominantly on external appearances. Sizing is based on count per 13.6 kg (30 lb.) container, most typically 4 or 5, and occasionally 6 melons per carton. High quality appearance is protected, in part, by packing with a partition to protect melons from bruising, compression and scuffing. 
To properly store Honeydew: 45- 50 degrees F  Storage life is typically 12-15 days at 45 degrees F with up to 21 days attainable. 

Source authorities vary in the reported optimum storage and shipping temperatures for honeydew melons. Most recommendations use  45 degree F and 85-90% R.H. as the optimum handling conditions. In general, if melons are ripe or pretreated with ethylene at 100 ppm for 24 hr, trade recommendations for short-term storage and shipping are often range from 36.5 - 41 degrees F. Extended holding at these temperatures will induce chilling injury, rapidly evident after transfer to typical retail display temperature 

Here are some good links for Honeydew information:
http://newcrop.hort.purdue.edu/rhodcv/hort410/muskm/mu00002.htm

http://www.wegmans.com/kitchen/ingredients/produce/fruit/honeydew.html

http://postharvest.ucdavis.edu/Produce/ProduceFacts/Fruit/honeydew.html