Having looked at text, and seen its original meaning, it naturally follows that we should examine the word weave. Considering the importance and long history of the activity through the centuries, it is not surprising that this is an ancient word.
We find an early reference to it as the verb wefan in the 12th century, in the Homilies of Ælfric:
Ge lufigeð þiss lif, on þan þe ge mid geswynce wuniged … þu grintst, & bacst, þu wefst, & wæden teolest, & earfoðlice wast ealre þinre neoda getæl.
(You loved this life, and then you lived with working … you ground, and baked, you wove, and would toil, and with difficulty were all your needs counted.)
You can see it here as wefst, in the second person singular verb form, and it had much the same meaning as it does today.
The Old English form came from Proto-Germanic *weban. We can still see it today in other Germanic languages too: weben in German, weven in Dutch, væve in Danish, väven in Swedish, veve in Norwegian.
However, Proto-Germanic didn’t just invent the term. Words in other languages indicate an earlier source – Proto Indo European *webh-, meaning “to weave”. Your first thought on seeing that might be another English word; web. And indeed, you would be correct. Web comes from the same root, and its Old English form webb meant “woven fabric”. After all, what is the most basic woven thing in the world around us, if not a web, a spider’s web? A natural association.
Let’s take a look at other, non-Germanic, languages that trace their terms back to this same root. Firstly, in Welsh, the word gwehyddu means “weave”, and is likely from *webh-, not to mention the modern use of gwefan to mean “website”.
In Albanian, the word vegjë means “loom” or “weaving frame”.
And in Greek we find the word υφαίνω (ifaino), which means “weave”. This also developed from *webh-, via the evolved stem *ubh-, formed from the former, with a consonant shift to a /f/ sound, similar to the English shift to a /v/ sound.