Silent prayer originated with the Quakers, and has been a part of Quaker prayer rituals since their founding in the mid 17th century. The first recorded instance of it being in response to a death, however, comes from Portugal in February 1912 after the death of José Maria da Silva Paranhos, the baron of Rio Branco, Brazil.

There are two prevailing theories as to the popularization and secularization of the practice. The first is from Cape Town, South Africa, in which a moment of silence would be held in honor of the fallen soldiers of the Great War. It was held every day until the armistice on November 11th of 1918.

The second is that it was proposed by Australian journalist Edward Honey, who, in 1919, wrote a letter to a British newspaper suggesting that the anniversary of the armistice should be greeted with five minutes of silence.

In either case, the idea was brought before King George V by MP Sir Percy Fitzpatrick and it became official policy on November 17th, 1919 to have 2 minutes of silence on the eleventh hour of the eleventh day of the eleventh month every year.

As to when this exended to sports it's difficult to say, but almost assuredly came out of this two minutes silence that was official policy across the British Empire. If a sports historian (Indy pls) could come in and clarify the first instance of a moment of silence in sports, that'd be great.