This word appears in the phrase no aer i eneth lín “hallowed by thy name”, the second line of Ae Adar Nín, Tolkien’s Sindarin translation of the Lord’s Prayer. It is untranslated, but in context clearly means “*hallowed, holy”. It is apparently the Sindarin cognate of Q. airë¹ of similar meaning, which appears in the Quenya translation of the same prayer: Q. Átaremma.
Possible Etymology: David Salo suggested that aer may be a lenited form of the Sindarin adjective gaer¹ (GS/231). Although N. gaer was glossed “dreadful” in The Etymologies (Ety/GÁYAS), in later writings Tolkien derived Q. airë “holy” from a similar root √GAY(AR) “awe, dread” (PM/363). The sense “dread” did not necessarily connote terribleness, but could also imply awe and respect. It was from this meaning of the root that words with the connotation “holy” arose, at least in Quenya (PM/363).
Bill Welden instead connected S. aer “holy” with the word aerlinn in the phrase Aerlinn in Edhil o Imladris (RGEO/62). This phrase is untranslated, but it is widely believed that aerlinn means “*holy song” (VT44/24). Since aer cannot be lenited in aerlinn, this indicates that aer is the normal form of the word. In other writings, Tolkien derived Q. airë from primitive ✶airē “holiness, sanctity” (PE17/149), which could also be the origin of S. aer. That is the theory used here.
It is possible that this word appears in the name Aerandir, but that is rather speculative.
Conceptual Development: In the Gnomish Lexicon from the 1910s, there was a word G. air(in) “holy, sacred” (GL/18) marked archaic (†); this is probably the earliest precursor to S. aer. The form air reappeared in the Gnomish Lexicon slips, besides a new form eirin “holy” (PE13/113), reflecting Tolkien’s ongoing vacillation on the development of the diphthong ai.