button connected to mobile phone

Hi, guys!
I am sorry, if this is related to another topic.

I am actually web programmer and do not know anything about electronics, arduino and so on. However, I want to create a small project with my 2nd grade kid. The idea is more complicated, but for now what I need is to have:
a) button
b) mobile phone somehow is connected to button (for instance, via bluetooth)
c) mobile phone application counts how many times button was pressed.

Having this in mind I started googling for 'something' (actually, not quite sure for 'what') and found out Arduino. So basically my question is:

  • whether we can do this with Arduino or I am totally wrong?
  • if yes, then what are your suggestions - what should I start with? what to buy?
  • if no, maybe you can point out some other usefull tips for "eletronics dummy", "diy kits related to this project", anything else?