Why exactly is the NHL Stanley cup called stanley cup? Was it named after someone named Stanley in the NHL?

It literally was Lord Stanley's Cup, which it is still often called. He donated it to the winners of the top national hockey tournament at the time. The Cup is much older than the NHL itself.

Others herein have explained the name of the cup.  For the record, Allen Stanley was a defenseman for the Toronto Maple Leafs in the 1950's-'60's.  He was on the cup winners in '62, '63, '64, & '67.  For young Leaf fans in Windsor, Ontario, 1962 was a great year as not only did the Leafs win the Stanley Cup, but the Windsor Bulldogs won the Allan Cup, awarded to the champion of Senior League play.     

It was named after Lord Stanley of Preston who donated it.

It was named after the British nobleman Lord Stanley of Preston, who was Governor General of Canada at the time.  Stanley commissioned the trophy, originally the Dominion Hockey Challenge Cup, in 1892 when he was still governor, to be given each year to the top amateur club in Canada. The trophy was re-named in his honor and was given to amateur clubs exclusively until 1906, when professional clubs could compete for it. In 1915, the top two professional leagues agreed to play for the Stanley Cup.  The NHL came about in 1917, but the Cup did not become the de facto championship trophy until 1926.  Contrary to public perception, the NHL does not own the trophy. The trophy is held in trust by two Canadian trustees, who have a long-standing agreement with the NHL for the leagues exclusive right to use it.

The bowl (the top) was donated by Lord Stanley of Preston as a trophy for the best amateur hockey team in Canada.

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