There a leak in our first floor bathroom ceiling. This leak happens only when it rains. Above the first floor bathroom is the 2nd floor bathroom. There is no water on the 2nd floor bathroom. The toilet on the 2nd floor flushes fine. The 2nd floor bathroom is part of a room addition. I had contractor come out and he tells me that it is because the siding on the outside of the room addition is loose and needs to be taken out and reinstalled. Also, he states that I have to get a gutter for the room addition overhang.
Does this sound right?

Sounds reasonable to me. If it doesn't leak on a daily basis, nor when any of the plumbing is used on the second floor... then that isn't likely the cause... and thank your lucky stars for that! That is a much messier clean up. Was the addition added recently, and was it done to code with permits? If so, go back to that same contractor and have him come out and fix it! If that answer is no... you did it yourself... then it sounds like you didn't properly seal it for water tightness.

Is it the rainy season where you are NOW? If not, take a hose/sprinkler up on the roof and water the entire addition. Have someone listen on the inside for trickling water inside the wall to locate the initial breach. Once found... inspect the entire edge of the roofline above the addition as well as where it sits on the 1st floor. Is there a noticeable flashing and/or water barrier along those edges... and caulking to seal the entire seam?

If I were you, I wouldn't seal the hole up with out removing either the inside or outside wall. You need to get a look at the damage on the inside... replace any wood rot or decay, check for termites and other pests, and worse case scenario... mold. Some molds can make you very sick, and in some lead to death. As far as the gutter along the overhang... it's a good idea. This will work to divert the majority of the water rolling down the 2 floor roof, and keep it from pouring down on the first floor roof and causing more wear and tear. This will probably be the cheapest part of the whole project. Good Luck.