This year I planted honeydew for the first time. The plant is very healthy but the odd thing is every melon comes out in a funny shape!

Veggie-T

Just a guess, but they may have been attacked by aphids just prior to the fruit beginning to develop. This will often cause weird shapes and abnormal growth to both leaves and fruit.
A tell tale sign that aphids are going after your plant are gnarly mishapend leaves. If that is the case you can use either Sevin or Malition to rid them from your plants. Just be aware of the time to harvest statement that is on the label. This will tell you how far before picking you can do an application.

Hope this helps

Mike