Hebrew ḥēn means favour or grace, and the name built on it — Ḥannah — has had a career matched by very few. Through Greek and Latin it became Anna, and from there Ana, Anne, Ann, Annie, Hanna, Hana, Anya, Aine and several dozen more, spread across every language Christianity reached. It has been in continuous use for something close to three thousand years, and it is probably the single most successful girls' name in history.
Greek cháris gives the Charites, the three Graces, and a family of names in Greek and beyond. Latin grātia — from a root meaning to praise or welcome — produced Grazia, Gracia and the English Grace, which arrived as a Puritan virtue name in the sixteenth century rather than as a classical inheritance.
Arabic and Persian approach the idea through elegance and delicacy rather than through favour: Persian nāz and its many compounds describe a quality of graceful bearing, sometimes with a hint of playfulness that the English word does not carry.
The theological sense — grace as something given rather than earned — sits behind most of the European names here, and it is worth knowing that it is present even where the name has become entirely secular in use.
The same root, across a whole language family
These are attested shared morphemes — not coincidences of spelling. Where we are not confident a root is shared, we say nothing and simply list the names.