Husseini didn't tell Hitler to exterminate the Jews, and the issue wasn't discussed at all in their meeting. The full record of the conversation is available in Documents on German Foreign Policy 1918-1945, Series D, Vol XIII, the record of their conversation appears on pages 881-885.
A second source containing the full book, just in case.
An article in The Times of Israel with only the transcript of the conversation.
While both mention the Jews as the natural enemies of both the German and the Arab people, they don't talk about the solution to "The Jewish Problem", nor does the Mufti offer any advice on how to handle the Jews.
Hitler does say that:
The Fuhrer then made the following statement to the Mufti, enjoining him to lock it in the uttermost depths of his heart:
- He (the Fuhrer) would carry on the battle to the total destruction of the Judeo-Communist empire in Europe.
- At some moment which was impossible to set exactly today but which in any event was not distant, the German armies would in the course of this struggle reach the southern exit from Caucasia.
- As soon as this had happened, the Fuhrer would on his own give the Arab world the assurance that its hour of liberation had arrived. Germany’s objective would then be solely the destruction of the Jewish element residing in the Arab sphere under the protection of British power. In that hour the Mufti would be the most authoritative spokesman for the Arab world. It would then be his task to set off the Arab operations, which he had secretly prepared. When that time had come, Germany could also be indifferent to French reaction to such a declaration.
Moreover, Hitler assured the mufti that he was against creating a Jewish homeland in Israel:
The Fuhrer replied that Germany’s fundamental attitude on these questions, as the Mufti himself had already stated, was clear. Germany stood for uncompromising war against the Jews. That naturally included active opposition to the Jewish national home in Palestine, which was nothing other than a center, in the form of a state, for the exercise of destructive influence by Jewish interests. Germany was also aware that the assertion that the Jews were carrying out the functions of economic pioneers in Palestine was a lie. The work there was done only by the Arabs, not by the Jews. Germany was resolved, step by step, to ask one European nation after the other to solve its Jewish problem, and at the proper time to direct a similar appeal to non-European nations as well.
In the end of the conversation the mufti wanted a secret agreement of some kind and Hitler assured him that that is what had just happened:
The Grand Mufti replied that it was his view that everything would come to pass just as the Fuhrer had indicated. He was fully reassured and satisfied by the words which he had heard form the Chief of the German State. He asked, however, whether it would not be possible, secretly at least, to enter into an agreement with Germany of the kind he had just outlined for the Fuhrer.
The Fuhrer replied that he had just now given the Grand Mufti precisely that confidential declaration.
The mufti did play a role in WWII, and and he did try to stop deportation of Jews out of Romania and Bulgaria, at least one instance he was successful, and 4000 Jews were sent to Auschwitz instead.
A Jerusalem Post article cite Dr. Efraim Zuroff from the Simon Wiesenthal Center and Dr. Rafael Medoff, director of the David S. Wyman Institute for Holocaust Studies in Washington:
While Netanyahu was incorrect in his assertions, it is likely that his confusion was based on the mufti’s very real role in lobbying against proposals to bring Jews out of Romania and Bulgaria during the war, Zuroff said.
“The mufti tried very hard to stop it,” Zuroff said. “He intervened to make sure it wouldn’t happen and that intervention apparently halted or stopped those operations.”
One such incident involved the mufti “persuading the Germans to cancel a 1943 prisoner exchange that would have sent 4,000 Jewish refugee children to Palestine,” said Dr. Rafael Medoff, director of the David S. Wyman Institute for Holocaust Studies in Washington.
“As a result of the mufti’s intervention, the children were sent to Auschwitz. There is ample evidence that he knew they would be murdered,” Medoff said.
Zuroff said that the involvement of the leader of the Palestinian national movement in the Holocaust is a matter of historical record. He cited the mufti’s propaganda work on behalf of the Third Reich and his role in recruiting Muslim troops to take part in the Holocaust.
“There is no doubt that the mufti was a zealous supporter of the Third Reich and that he hoped that the Nazis would implement the Final Solution in the Land of Israel, but Hitler did not need any convincing from the mufti or anyone else to launch the annihilation of European Jewry,” Zuroff said.
A Ynet article citing Prof. Meir Litvak from Tel Aviv University:
Professor Meir Litvak, who teaches at Tel Aviv University's Department of Middle Eastern History, ... "Husseini supported the extermination of the Jews, he tried to prevent rescuing of Jews, he recruited Arabs for the SS," said Litvak. "He was an abominable person, but this must not minimize the scale of Hitler's guilt."