P. Lloyd points out several interesting things.
How /eu/ went from hiatus to diphthong
- The vowel sequence /eu/ was extremely rare in Classical Latin, being found only in five native words which then disappeared from popular speech, and in a few learned Greek loanwords like Europa.
- These /e/ and /u/ were in hiatus, but later evolved into a descending diphthong, just like the other (much more common) /ai/ and /au/.
- By the time of early Romance we find a few words with this /eu/ dipthong (phonetically that would have been [ɛu̯]): meu, Deus, jūdaeu, *eo < ego.
How /eu/ changed into /jeu/ ~ /jou/
- The regular development of stressed [ɛ] was to /je/ and there is no reason to think that /eu/ would not have changed regularly into /jeu/. This is in fact what we find in some dialects of Western Asturian, where the reflexes of meu, Deus and ego are /mjeu/, /djeus/ and /jeu/. It is extremely likely that closely related Castilian would also have followed the same regular route.
- Why did /jeu/ later change into /jou/? Lloyd postulates analogy. The diphthong was extremely rare. It appeared most often in the word for "I" (presumably /jeu/), and this was often found right next to two verb forms that contained the diphthong /ou/: vou (< vadō, modern Spanish voy, "I go") and sou (< su(m), modern Spanish soy, "I am"). Rhyme often motivates sound change, either towards or away from it.
- An additional common verb form, the third person preterite of dar "to give", is reported as /djeu/ at one stage. This one might have been influenced by the regular ending /o/ (modern Spanish amar > amó, temer > temió, etc.), giving then /djou/.
- Once these three extremely common short words had shifted /eu/ to /ou/, the others followed suit by analogy.
- Of the four words from Late Latin with phonetic [ɛu̯] that we began with, two have evolved into an ascending diphthong (modern yo and Dios, forgetting for the time being the change of consonantal /j/ into a fricative in most modern dialects of Spanish), while two have restored a hiatus (mío and judío).
- Why is that? Lloyd again postulates regularization by analogy. Mío has a feminine counterpart, mía < Latin mea, where the hiatus was never resolved. If /mjo(u)/ existed, it would have contrasted with /mi.a/, a very odd irregularity; therefore it was "brought back" as /mi.o/.
- Likewise, final-stressed judió must have sounded rather weird, and there was a regular feminine word, judía (< Latin jūdaea), which hadn't shifted stress; therefore judío had a hiatus restored as well.
- In case this seems farfetched, Lloyd states that there are indeed attested medieval reflexes of those two words as mió and judió.
So it seems that Dios was always like that. We can't actually be sure whether at some moment, somewhere, it had a variant pronounced like Díos, but that's not apparent.
ADDENDA: Portuguese, unlike Spanish, preserves the old descending diphthong /eu/ in the words eu, meu, Deus, judeu. It could very well be that the preservation of Vulgar Latin's lax vowels (open e and o) has something to do with this, if the route by which /eu/ changed into /jo/ was indeed through /jeu/ > /jou/, as Lloyd says. That is, where Spanish changed stressed open e to ie, Portuguese did no such thing, so for example Latin petra > Port. pedra, so the e in Deus never acquired a glide in Portuguese.
Regarding the expression a brios rogando, I don't think the stress pattern needed to be the same in bríos and *Díos for the rhyme to work, if the phrase was said in one breath (as fixed idioms tend to be) and the main stress was on rogando.