From eu (good) + charis (grace/favor), eucharisteo is to give thanks — literally to declare that good grace has been received. It is the verb from which the Church took 'Eucharist' as its name for the Lord's Supper, as Jesus gave thanks (eucharisteo) over the bread and cup.
Paul builds a theology of radical thankfulness around this word. 'In everything give thanks' (1 Thess 5:18) and 'be thankful' (Col 3:15) are not optional extras but marks of Spirit-filled life. Interestingly, ingratitude is the root of the Gentile moral collapse in Romans 1:21 — they 'did not give thanks.' Thanksgiving acknowledges God as the giver of all good gifts, rightly orienting the self in relation to God.