According to Hans-Peter Vietze, Chandmani means Jewel or precious thing. However, I am still not sure wether in this case Chandmani Uul is a toponym nor not. There seems to be a Chandmani uul near Ulaangom (source: a picture book about Uvs, by G. Tsatsralt (or similar), Uul Us Nutag (or similar), UB 2005), but it didn't look very big on the picture. Of course that doesn't mean it can't be significant for other reasons. 
Baabar uses the genitive forms in Mongolchuud: Nuudel Suudal (paperback), UB 2006, vol.3 (Togloomyn Bugd Nairamdakh Uls),p.122 (=end of chapter 12). The other Aimags given are Khan Tashir Uulyn, Tsetserleg Mandal Uulyn, Bogd Khan Uulyn, Khan Khentii Uulyn aimag. At least the latter two seem to be named after real mountains.