Look at the Rav Sapurna Breishis (16,15) who says that the name Yishmoel was applicable because of both. Avraham davened for him whereas Hagar heard it from the Angel. It seems that Avraham called the name on his own because Hashem listened to his tefila Breishis (17,20).
As a side question, Rivkah was told, the older will serve the younger. Did Yaakov Avinu know this? The Maaseh Hashem explains the connection between Aisov asking to be served the lentil food and Yaakov Avinu wanting him to sell his birthrite. Yaakov Avinu said, if you, Aisov want me to serve you, I must be the older.