Latin 'to set a price on' — originally an appraisal, broadened to recognizing worth and then to gratitude.
To recognize the full worth of something; to be grateful for something; to increase in value.
From Late Latin 'appretiātus,' past participle of 'appretiāre' (to set a price on, to appraise), from Latin 'ad-' (to) + 'pretium' (price, value, reward). The original meaning was commercial — to determine the monetary value of something. The sense 'to recognize the worth of' (whether monetary or not) developed in the 17th century, and 'to feel grateful'
'Appreciate,' 'price,' 'precious,' 'praise,' and 'appraise' are all from Latin 'pretium' (price). When your house 'appreciates,' it literally 'gets priced up.' When you 'appreciate' a friend, you are etymologically 'putting a price on' their worth. And when something 'depreciates,' it 'de-prices' — loses value