Honeydews will become softer and juicier after picking but they will not grow sweeter, so it’s important to choose wisely. During my early honeydew days, many a fruit ended up in the trash because I mistakenly assumed they just needed time to ripe. There is a world of difference between a bland, pale honeydew and the deep green, juicy-sweet flesh of one that has been allowed to ripen on the vine. Look for melons that are slightly soft, and with a creamy yellow navel. Rock-hard melons will not be sweet. You should also detect a pleasant honey-like aroma. And, as with most fruits, the better the taste, the richer in nutrients.