From the earliest times, the Elves had dedicated a day of the week to the Moon. They named this day Isilya in Quenya, or Orithil in Sindarin. The Númenóreans adopted the Elves' system (with some adjustments) to their own use, and ultimately transmitted these to Middle-earth. Monendei ('Moon-day') is an archaic form in the Common Tongue of this name, and by the time of the War of the Ring, this would be worn down to the familiar 'Monday'.
Notes
1
In Appendix D to The Lord of the Rings, it is said that the old form Monendei was in use by the Shire-hobbits at least nine centuries before the end of the Third Age, which would place its use from approximately the year III 2100 or S.R.500. Over the following nine centuries, the name evolved and simplified, so that by the end of the Third Age the more usual form was simply 'Monday'.
2
Our only specific record of this usage is from the history of the Shire. Its use in Buckland or the Bree-land seems likely, but is not directly attested. For further discussion on this point, see footnote 2 to the entry for Sterrendei.