Serpent Eggs can be found by attacking Level 5 Lakes or higher.
There is a greater chance of acquiring a Serpent Egg in higher level Lakes. In other words, Level 10 Lakes have a higher probability of dropping a Serpent Egg than Level 10 Lakes.
Serpent Eggs can appear in Lakes you have already defeated, so feel free to attack the same Lakes multiple times to increase your chances of finding an Egg.
Using a Wraith Dragon to find this Egg will significantly increase the odds of finding it.
You can also get a Serpent Egg by purchasing a Serpent Egg Item in the Shop.