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I found a source which gives the PIE origin: http://www.utexas.edu/cola/centers/lrc/ielex/X/P1589.html But it only lists Slavic reflexes. Are there related words in Persian or Sanskrit?
Wikipedia claims the name of the Indian goddess Radha is related https://en.wikipedia.org/wiki/Radha#Etymology, but gives no source for this claim.
As so often, Wikipedia is completely wrong about this. The Sanskrit name Rādhā cannot possibly be related to Persian rād “generous”. Indo-Iranian post-vocalic dh becomes y in Persian, not d.
The etymology of Slavic root is also unclear. Vasmer writes that Greek sources give the following versions of the names of Slavic princes: ᾽Αρδάγαστος, ᾽Αρδαμέρι, which are evidently the versions of Slavic names Radogostь "happy with guests" and Radoměrь respectively.
This hints the root had something before the r-. Some link it to Greek root ἔρως "love" which in turn also has obscure etymology.
If this theory wrong, then possible cognates (according to Vasmer) are Anglo-Saxon rǿtu "joy" and Old Islandic rǿtask.
Note by the way, that PIE prohibited roots starting with r- altogether, a rule about which Vasmer (and Pokorny) could not know.
