No one knows for sure why Alfred Nobel wanted the Peace Prize in particular to be awarded by a Norwegian committee - or what prompted him to include Norway in the Nobel Prize proceedings at all.
In his article The Nobel Peace Prize, 1901-2000, former secretary of the Norwegian Nobel Committee and director of the Nobel Institute, Geir Lundestad, summarizes the most frequent educated guesses about Alfred Nobel's possible motivations for asking the Norwegian Storting to select members for the Nobel Peace Prize awarding committee.
"Nobel left no explanation as to why the prize for peace was to be awarded by a Norwegian committee while the other four prizes were to be handled by Swedish committees. On this point, therefore, we are dealing only with educated inferences. These are some of the most likely ones: Nobel, who lived most of his life abroad and who wrote his will at the Swedish-Norwegian Club in Paris, may have been influenced by the fact that, until 1905, Norway was in union with Sweden. Since the scientific prizes were to be awarded by the most competent, i.e. Swedish, committees at least the remaining prize for peace ought to be awarded by a Norwegian committee. Nobel may have been aware of the strong interest of the Norwegian Storting (Parliament) in the peaceful solution of international disputes in the 1890s. He might have in fact, considered Norway a more peace-oriented and more democratic country than Sweden. Finally, Nobel may have been influenced by his admiration for Norwegian fiction, particularly by the author Bjørnstjerne Bjørnson, who was a well-known peace activist in the 1890s. Or it may have been a combination of all these factors." (Source: nobelprize.org)