The word 'been' — the past participle of 'to be' — comes from the third of the three PIE roots that were fused together to form the English verb 'to be.' While 'am/is' descend from *h₁es- (to exist) and 'was/were' from *h₂wes- (to dwell), 'be/been/being' descend from PIE *bʰuH- (to grow, to become, to come into being).
The Old English infinitive 'beon' (to be, to become) came from Proto-Germanic *beuną, from PIE *bʰuH-. In Old English, 'beon' and 'wesan' coexisted as two separate verbs meaning 'to be,' with a subtle distinction: 'wesan' referred to present states, while 'beon' had a future or habitual sense — 'beon' meant 'to come to be' or 'to regularly be.' This distinction was gradually lost as the forms merged into the single modern paradigm.
The PIE root *bʰuH- meant 'to grow' or 'to become,' and this dynamic, processual sense survives in many of its descendants. Sanskrit 'bhavati' means 'he becomes, he comes into being.' Greek 'phúō' (φύω) means 'I grow, I bring forth' — and from it comes 'phúsis' (φύσις, nature, literally 'growth'), the source of English 'physics,' 'physical,' and 'physiology.' The connection between 'been' and 'physics' is real: both descend from the same PIE root about growth and becoming.
Within Germanic, the root *bʰuH- produced a rich family. German 'bauen' (to build) comes from *bʰuH- through the sense 'to make something grow/become.' 'Bauer' (farmer, peasant) is literally 'one who makes things grow.' English 'build' has the same origin — a causative form meaning 'to cause to become.' 'Booth' comes from Old Norse
The most surprising member of this family is 'husband.' Old Norse 'húsbóndi' means 'master of the house,' from 'hús' (house) + 'bóndi' (householder, dweller), from 'búa' (to dwell), from *bʰuH-. A husband was originally a house-dweller, a settled farmer — the word had nothing to do with marriage until later. 'Husbandry' (farming, careful management) preserves the older sense.
'Bower' (a leafy shelter, a lady's private chamber) comes from Old English 'búr' (dwelling, chamber), also from *bʰuH-. And 'neighbor' is 'nigh-bower' — 'the near dweller,' the one whose dwelling is close to yours.
The pronunciation of 'been' varies notably between dialects: British English typically uses /biːn/ (rhyming with 'seen'), while American English commonly uses /bɪn/ (rhyming with 'bin'). Both pronunciations are historically legitimate descendants of Old English 'bēon,' reflecting different paths of vowel development across the Atlantic.