kuriakē(i) (LSJ) (from κυριακῇ ἡμέρᾳ) is an adjectival form of kurios, "lord", which could be rendered "lordly" (on analogy of "royal" = "kingly", roughly!). As the adjective "royal" indicates something belonging to the monarch ("the royal palace"), so kuriakos indicates something belonging to the "lord".
Rev 1:10 uses it with day: "on the lordly day" (= bad translation! just to attempt to clarify) + see other phrases at link, above; whereas "Sarah's Day = Day of Sarah" is like "Lord's Day = Day of Lord" from OT.
Some other early Christian writings use the Rev 1:10 phrase. In Didache 14:1, for example:
On the Lord's Day of the Lord come together, break bread and hold Eucharist, after confessing your transgressions that your offering may be pure
Which precise day is in mind of these options (first day? Sabbath day? Easter Day?) is not specified. However, one or two of the early Christian apocryphal writings are explicit aboout which day this is, e.g. Acts of Peter, in the prologue [scroll down to second line of
I. THE COPTIC FRAGMENT]:
On the first day of the week, that is, on the Lord's day...
These are from the second century, however, and might be deemed to be too late to determine the usage of the phrase in Revelation 1:10.
Some of these references and themes are touched on already by James Moffat in the Expositor's Greek Testament (Hodder & Stoughton, 1897), vol. 5, p. 342. For a more modern treatment see (among many options), David Aune, Revelation 1-5 (Word; Dallas, 1997), p. 84.
Just about all the data and assessment one could want on this are assembled in the article by R. Stefanovic, "'The Lord's Day' of Revelation 1:10 in the Current Debate", Andrews University Seminary Studies 49/2 (2011) 261-284.