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faoilinn f. [ˈfɯ:liɲ] ‘gravel bank or sandbank’; the range of forms of this word theoretically includes (with non-palatalised or palatalised ending) faoileann, faoilinn, (without initial f-) aoireann, aoirinn and (with non-palatalised medial consonants) faolann, faolainn, aorann, aorainn.
The word is relatively common as a place-name element on the west of Scotland:
(1) (South Uist) Peighinn nan Aoireann occurs on the west coast of South Uist, NF747350 (Taylor 2011, s.v. Peninerine).
(2) (Arran) An Aoirinn [ə ɴöːr’in’] 
Which Robertson (in King 2019, 412) spells an Éirinn, i.e. with a stressed vowel close to [eː] or [øː].
For early documentary forms of Rubha na h‑Aoireann, see Fraser 1999, 149.
Robertson: a’ Chlacha͝or̀ainn (for medial [rʲ], Robertson uses an r with a grave accent).
This spelling suggests a stressed vowel [əː] or [ɤː] and a connection with fadhail, q.v.
Unless [əː] is in error for [ɯː] here; cf. AFB˄, which gives fadhlainn /fɤːʟɪɴʲ/ ‘raised beach’ for locations in Lewis, Skye and mainland Scotland.
But given as An Fhaolainn in MacDonald et al. 2018, 119.
Henderson (ibid., 140–42) suggests the word is based on ON fjara f. ‘ebb-tide; foreshore, beach’ + suffixed article (but see below), although ON fjaran seems phonetically unlikely to be behind the Gaelic forms. MacBain (1896) takes faoilinn to be from ON vaðlinn, from vaðill m. ‘wading place, shallow water’ + suffixed article, but neither the phonetics nor semantics suit. Given the range of forms in Gaelic, McDonald (2009, 353) suggests both ON fjara and ON vaðill might be involved to some extent but concludes the loan is unlikely. MacBain (1922, 174, 350–51) takes the specific element in Peighinn an aoirinn (Peighinn nan Aoireann, South Uist), Rhu na h‑aoirinn (Rubha na h‑Aoireann, Kintyre) and Eilean na h‑aoirinn (Eilean na h‑Aoirinn, Jura) to be for aifreann m. [ˈafɾʲəᵰ̪], aifhreann [ˈaiɾʲəᵰ̪] ‘mass’, but the stressed vowel is substantially different. Gillies (ibid., 187) claims aoirean [sic 
It is assumed Gillies means aoireann.
Senses given tend to be location specific: ‘level raised beach, not high above the sea, not necessarily grassless, of fair extent, e.g. at least eighty to a hundred yards long; in some parts a sandy spit, a sandy part of the shore where rocks abound on either side’ (CG VI, 70: faoilinn, faoileann); ‘a ridge above a sandy beach with a marsh on its landward side; sometimes awash during spring-tides’ (MacDonald 1946, 24); more generally it is described as a ‘raised beach; stony beach; field or stony place by the shore’ (e.g. Faclan bhon t-Sluagh˄; Campbell 1969, 360; Taylor ibid., s.v. Faolin); MacKinnon (ibid.) points out that the aoireanns of South Argyll are or have been ‘ferries’, but Henderson (ibid., 141) notes this must be secondary.
A derivation from ON eyrr f. ‘gravel bank or sandbank’ is obviously attractive from a semantic point of view, but ON ey can normally be expected to yield [eː] or [ia] in Gaelic, e.g. (Lewis) ON *Streymanes > SG Srèiminis [ˈs̪t̪ɾẽːməˌniʃ], ON *Steypavatn > SG (Loch) Stiapabhat [ˈʃtʲiaʰpəˌvaʰt̪] (Cox 2022, 178). However, ON ey was raised by around 1050 from CSc. ø̨y [œy]. CSc. ø̨y may have approximated the Early Gaelic diphthong áe, aí, óe, oí, e.g. EG cáel > (Lewis) caol [kʰɯːɫ̪] (Oftedal 1956, 82: /kɯːʟ/), (Arran) caoil [køːl], [ke̩ːl] (Holmer 1957, 77: [kEːl’]).
For a comparison of dialectal reflexes of EG áe in Scottish Gaelic, see SGDS Item 147: caol.
CSc. ø̨yrr, or ø̨yri acc./dat., borrowed into Gaelic as a feminine n-stem, might yield SG (nom.) aoir(e), (acc.) aoirinn, (gen.) aoireann, (dat.) aoirinn, with acc./dat. or gen. forms later normalised as radical (nom./acc.) forms, cf. EG lecc f. ‘flat stone’ (SG leac) > SG leacann f. ‘slope etc.’, EG goḃae m. ‘smith’ (SG gobha) > EG goḃann ‘idem’ (SG gobhann, gobhainn). SG aoireann in the south might easily yield faoileann, faoilinn to the north, with prothetic f- (cf. farspag for arspag, q.v.) and interchange of l for r (cf. coireach adj. (Lewis) [ˈkʰaɾʲɔx], [ˈkʰalɔx], ruig vb (Lewis) [liɡ̊ʲ] (in the phrase cha ruig ... a leas 
Although this involves morphemic substitution. For further examples of l > r, see Calder 1972, 67–68.
Who cites various suggested derivations: aoireann ‘foreshore, beach’, irionn ‘field’, na h‑aorainn ‘the place of worship’, earrann ‘section of land’, and [aoireann] ‘sandbank or shoal’.
In Tiree, SG *Aoir NL948477 seems to be a loan-name (rather than a loan-word) from CSc. *Ø̨yrr or *Ø̨yri acc./dat. (Holliday 2016, 246–47). Later ON eyrr is frequent in loan-names, e.g. in Lewis in SG Èire < ON *Eyri dat., SG Èire Solt < ON *Eyrin Sǫltu dat. and SG (Loch) Èirearaigh < ON *Eyrará (gen.) (Cox 2022, 702, 808).