To make the long story short –
Quran does not specifically say Ishmael.
The Bible says it was Isaac.
There are contradicting Islamic traditions on who actually was put up for sacrifice.
In the Bible it was clear that Isaac was the one.
Many times Muslims mentality of proving whether a belief is true or not, it must be stated catergorically in the scriptures. Example, if you believe that Jesus is God, where does in the Bible that Jesus says “worship me”?
Now, since Ishmael is not mentioned as the would-be son for sacrifice in the Quran but Isaac was mentioned in the Bible, the conclusion should be Isaac. I think this would make everybody happy.
Ishmael was not mentioned in the Quran.
Isaac was mentioned in the Bible. If we choose Isaac, it will fit Muslims logic and Christians too will have no problem since they believe that son was Isaac.
Reuben