淘's semantic compound is 氵 (“water”). But how does "water" relate to "naughty; mischievous"?
淘 has the meaning of wash
'washing' is related to water
The action of washing is rapid and energetic
淘氣 (mischievous) is mostly used to describe an unruly child and we identify a child as unruly when that child doesn't seem to be able to sit still for a second
淘 itself doesn't contain the meaning of naughty; mischievous
"淘气" was not a Mandarin word at first. It is from dialect.
-Modern Chinese Dictionary, 2nd Edition, The Commercial Press, 1983, Beijing
Some dialect word appeared a long time ago. Some of them use words that are not common in today's Mandarin. Probably when "淘气" appeared, it was not written like this. But, because there is not a dictionary for the dialect or people always write it wrongly, the original form of it disappeared, and then it became "淘气". If so, 氵has nothing to do with "naughty". We need to do lots of research to find out its etymology.
I live in a city in Jiangsu province called Yangzhou. In the Yangzhou dialect, the situation is similar. For example, the word "掯(kèn)" means "to hold down", but most people cannot write it, so they use pinyin or words that sound similar. Also, "小气" was "小器" at first, meaning "器量狭小".