The Godzone Dictionary: Of Favourite New Zealand Words and Phrases (2006) says:
The expression right as something has been used in English since medieval times, using a string of comparatives, such as trivet or ninepence. Right as rain emerged in the 19th century and took precedence over all the other forms, possibly because of its pleasing alliteration, and also possibly because rain is perceived as good, and causes growth.
But a book review of the 1955 Dictionary of Early English by Joseph T. Shipley in the June-July 1956 edition of The Crisis magazine says:
Few of us are aware that many commonly used words once had meanings, in many cases, quite the opposite as those now current. Right, in the phrase "as right as rain," originally meant straight in direction.
And The Wordsworth Dictionary of Idioms (1993) agrees. It says as right as rain is:
A pun on the original meaning of right = straight.
The Free Dictionary gives these meanings of right:
11. Straight; uncurved; direct: a right line.
2. In a straight line; directly: went right to school.