Hi Hilary. we can say 真ん中の子（mannaka no ko） in a conversation.

Just a trivia, in case there are children more than three ; 上から4番目の子、上から5番目の子、、、 Commonly we count from eldest one but we also say; 下から2番目の子、下から3番目の子、、、 The second one from the youngest. This might sound funny in English though :) Hope this may help.