I don’t know.
I’ll build on Sid Kemp’s answer, and to use what online resources tell me.
Nemesos is used by Sophronius of Jerusalem (7th century), Anna Komnene (11th century), and the Byzantine lists of bishops.
Lemesos is used by Leontios Machairas (15th century), and the vernacular Byzantine chronicles (15th century).
The Turkish forms are Limasol or Leymosun; the Armenian is Limasol.
Going through Excerpta cypria; materials for a history of Cyprus : Cobham, Claude Delaval, 1842-1915, tr : Free Download & Streaming : Internet Archive , we have:
- Limeszun (1180s-1190s)
- Lamezis (1211)
- Limeso (1395)
- Lymiso (1484)
- Lymesson (ca 1500)
- Limechon (1518)
- Nimesso (1533)
- Minesse, Minesson (1534)
- Limisso (1553; 1570; 1589; 1598; 1609; 1626)
- Lemiso (1656)
- Lemisco (1683)
- Lumesol (1757)
- Limesol (1758)
- Lemissol (1745) (saying the Greek form is Nemeson)
- Llimaso (before 1790)
Based on all this:
My initial assumption was that is was an Old French contortion of the name Lemesos, just as Nicosia is Old French trying to deal with Lefkosia. But what few old instances of the name I find in the Excerpta Cypria don’t mention a form ending in -l until well after the Ottoman conquest. So I suspect it’s Turkish.
But I don’t know, and will ask further.