What is the origin of the word "goodbye"?

I've heard that goodbye comes from God be with you. Is that true? If so how did it become good? Did goodbye always have the same meaning it has now?

The OED also says that this is a contracted form of "God be with you", and explains the God-to-good transition thusly (emphasis mine):

A contraction of the phrase God be with you (or ye); see GOD n. 8. The substitution of good- for God may have been due to association with such formulas of leave-taking as good day, good night, etc.

In a quote from 1659, we still have "God":

But mum for that, his strength will scarce supply His Back to the Balcona, so God b' wy.

And then by 1694, we have transitioned to "good":

He flings up his tail..and so bids us good-b'wy.

The Merriam-webster confirms what you are writing in your question:

Etymology: alteration of God be with you Date: circa 1580

Note (Common Errors in English Usage):

GOOD-BY/GOOD-BYE/GOODBY/GOODBYE

All of these spellings are legitimate; but if you want to go with the most popular one, it’s “goodbye.”
This spelling has the advantage of recalling the word’s origins in phrases like “God be with ye.”

There are only two forms of farewell: “Good-by” and “Good night.” Never say “Au revoir” unless you have been talking French, or are speaking to a French person. Never interlard your conversation with foreign words or phrases when you can possibly translate them into English; and the occasions when our mother tongue will not serve are extremely rare.

From: Chapter III. ‘Greetings’ in Etiquette By Emily Post. 1922

Wiktionary tells us that the hyphenated variant, good-by, and the solid compound, goodby, are dated which everyone will agree. It also offers a more comprehensive list of variants that might be of interest to some.

Godby, Godby'e, Godbwye, God b'w'y, God bwy yee, God buy you, God be wi' you [“God be with ye”] each a progressively shorter contraction of God be with you.

Variants Between 1400 and 1700

Gid be with you (1400-1499)
God be with thee (1400-1499)
God be with you (1500-1700)
God be with yee; God bwy ye; God bwy; (1576-1600)
God be with ye (1576-1650)
God b'wee; God b'wy; God b'w'you; God b'wi'you (1601- 1625)
God buy ye; God buy you; (1601-1625)
God buy (1601-1650)
God be w'you (1626-1650)
God-buy (1651-1675)
God by ye; God-buy (1651-1675)
God bi wi'you (1651-1675)
God b'w'y' (1676-1700)
Godbuy (1676-1700)

The above is but a clumsy sample, I recommend visitors and enthusiasts to visit the complete table in the link below. According to the author, who painstakingly traced the lexical history of goodbye, the term Good (it remained capitalised) first appeared in 1676-1700 in the forms of:

Good b'w'y, Good b'we; Good b'wy to ye; Good b'uy to you; Good by t'ye;
Good by t'yee; Good-by

Source: p99; Diachronic Pragmatics: Seven case studies in English
illocutionary development By Leslie K. Arnovick