It actually has a cute history behind the name. For a while it didn't even constitute a capital W and was just the war of 1812 (instead of The War of 1812) and the British referred to it as "the American war of 1812." source. Also, another term used at the time was "America's Second War of Independence." John C. Calhoun used this term in an effort to garner support for the war effort source. Over time though there names were dropped from common use and the name The War of 1812 stuck. So based on what I am seeing there was nothing significant in calling it that, but rather it was the name that lasted out.