Euchomai (εὔχομαι) means to pray, wish earnestly, or make a vow. It appears about 7 times in the New Testament and is related to the more common proseuchomai (to pray). While proseuchomai is the general word for conversational prayer, euchomai often carries the sense of a deep wish or solemn vow.
Paul uses euchomai for his most anguished spiritual expressions. In Acts 26:29, he says 'I pray [euchomai] to God that not only you but all who are listening to me today may become what I am.' In Romans 9:3, he writes: 'I could wish [euchomai] that I myself were cursed and cut off from Christ for the sake of my people.' This is the most extreme form of intercessory desire in Paul's letters — wishing damnation upon himself for the salvation of Israel. It echoes Moses' intercession in Exodus 32:32. True intercession costs something — it desires others' salvation more than one's own comfort.