물찌똥 has two meanings. One is "loose stool," and the other is "splashing waterdrops." If you were to drop the 찌 in 물찌똥, you would get 물똥, which still means "splashing water," but no longer means "loose stool." That seems to imply that 찌 has a kind of "shitty" meaning in it, but I cannot find anything in the dictionary.
There is the word 꼴찌, which means "the last" or "the bottom," but I do not see any relationship with the 찌 in 물찌똥, unless you want to focus on the word, "bottom." There is also the word 찌개, which means "pot stew," but I cringe at the thought that that might somehow be related. Then there is the word 찌꺼기, which is kind of interesting since it means "dregs," "remains," "leftovers," or "waste," but even with 찌꺼기, I think I am still grasping for straws. Therefore, I guess I just have to give up on my search for the meaning of 찌.
Just remember that if you want to talk about "a water fight," you should use 물똥싸움, not 물찌똥싸움. However, I do think I remember seeing a 물찌똥싸움 is the movie, "Harold and Kumar Go to White Castle."