*Drags himself out of bed and immediately pounces upon the AS dictionary*
"Die" is the definite feminine article (nominative&accusative). It's actually not a huge coincidence because in AS it was þe (then a demonstrative pronoun, which, as I understand it, gradually replaced the definite article but has remnants, consider "that").
I think you're pretty much right. The table for the definite article is:
Ma. Neut. Fem. Plur.
Nom: se þæt seo þa
Acc: þone þæt þa þa
Gen: þæs þæs þære þara
Dat: þæm þæm þære þæm
As you can see, "Die" (which is the German Fem Nom and Fem Acc, as well as the Plural Nom and Fem, I think) tends to crop up in Anglo Saxon as "þa", which makes sense since they resemble each other. The exception is the Anglo Saxon Fem Nom, which is "seo"; this is the feminine equivalent to the masculine form "se", which means "the".
Now, about death:
The Anglo Saxons had a word for death: "deaþ". It's the same word. They also had a word for dead: "dead". Again, the same word. Their words for dying are less similar, however. The most straightforward word they have for "die" is "deadan", ie. "to dead"; they also use "steorfan" and "sweltan" which essentially translate as "to die", perhaps with an emphasis on "was killed" - although that might simply be because most deaths in AS literature are violent ones.
The other words they typically use for death are "gefaran" and "forðferan", which translate as "to travel" and "to go forth". They liked to allude to death poetically rather than simply say "he died". A lovely example of this is The Battle of Maldon where Wulfmær's death is described: "Wund wearð Wulfmær, wælræste geceas", "Wulfmær became wounded, he chose/found a slaughter-bed." (l. 113)
So where does "die" come from? Well, if we turn to our wonderful friend the OED, we get:
Etymology: Early Middle English dēȝen, dēghen, corresponding to Old Norse deyja (originally døyja, Old Swedish and Old Danish döia, Danish döe, Swedish dö), Old Frisian deia, deja, Old Saxon dóian, Old High German touwan, Middle High German töuwen; these represent an Old Germanic strong verb of the 6th class *daw-j-an, past tense dôw, past participle dawan-, the strong inflections being retained in Old Norse (dó- < *dów, dáinn < *dawans). In the other languages and in English a regular weak verb. No instance of the word is known in Old English literature (its sense being expressed by steorfan, sweltan, or the periphrastic wesan déad, past tense wæs déad: see dead adj. 1d) hence it is generally held to have been early lost in Old English (as in Gothic, and as subsequently in all the continental West Germanic languages), and re-adopted in late Old English or early Middle English from Norse; but some think that the facts point rather to the preservation of an Old English díegan, dégan, in some dialect; the word appears to have been in general use from the 12th cent., even in the s.w. dialects (see Napier in Hist. Holy Rood, E.E.T.S., 1894). The Middle English dēȝen, dēghen came regularly down to 1500 as deye, which was retained in the North as dey, dē, dee (still current from Lancashire to Scotland); but in standard English dēghe was in 14th cent. (in conformity with the common phonetic history of Old English eh, eah, eoh, as in dye, eye, fly, high, lie, nigh, thigh, etc.) narrowed to diȝe, dighe, whence the later dye, die.
So the full answer is that nobody knows for sure, but it seems likely that it was a word of Germanic origin which somehow didn't make it into AS but was later adopted.
