As a foreigner trying to open a bank account in Hungary, what arguments could convince a bank to approve it?

European bankers have become very strict towards transient bank accounts (which is that you are describing). No matter how noble your intentions are, you cannot convince the bank. This is a policy decision that probably took many months before it was decided upon and put into play. Policy won't change for you (or anyone else) because you happen to be wearing a nice suit and a smile today.

If you're American, things can be even more difficult because of FACTA (See: Foreign Account Tax Compliance Act)

Since you have not defined the purpose, I suspect, you know the reason why they are denying you banking.