It was when Arjun mentioned " My name is Arjun and I don't need anyone's help" and the precap in which Aakash says Arjun cannot be anyone's friend and no one can make Arjun their friend I got the symbolism in his name. Why his name is Arjun in first place.
So here goes the symbolism of that name. Arjun according to Mahabharth is a great warrior. He is a fighter, archer par excellence and in every way is the main protoganist of Mahabharath. But irrespective of being the great warrior his greatness would not have been acknowledged if not for one person. His friend. Krishna. Though incarnation of Lord Vishnu Arjun-Krishna relationship is treated as that of friendship as far as MB goes.
It is Krishna who protects him, supports him in his hour of need in the war. Being his friend's charioteer Krishna decides how Arjun will fight keeping his safety upper most. There are many incidents in which Krishna stands by Arjun. From the beginning when Arjun is not able to fight the war motivates him, plays with sun on 14th day to save Arjun, does not take Arjun to a head on battle with Karna till Karna has Shakthi, saves him from Nagastra being the key ones. So if Arjun had won the war Krishna had played an important role in it. In other words Arjun's trust in Krishna and their friendship was the key factor in the final victory.
Here also Arjun believes he can do it alone. But no. if Arjun is gonna reach the peak most probably it will be because of Anjali and Aakash. In other words Arjun has to start trusting, he has to be someone's friend and someone has to be his friend if he needs to accomplish his mission. That's why his name is Arjun. The man who thinks he is capable but unfortunately he is not. He needs love trust and respect from others to reach the glory. Sooner he accepts that better for him.