Now, there are two theories to this:
First, in ancient times, when agriculture was the primary mode of sustenance, rodents were one of the biggest obstacles to prosperity–as they are for farmers even today. Destroying crops, eating stored grains are all in a day’s work for a rodent. Lord Ganesha, in having a mouse/rat as his vehicle, is symbolically shown to have conquered this pest, thus staying true to his name and work as the destroyer of obstacles, according to biodiversity of India.
Second, Lord Ganesha was able to reach all the nooks and crannies of the world–and do his duty as the destroyer of obstacles–because his vehicle was a mouse–because mice/rats can slither through tiny holes and narrow pathways.