Is 'Ilana إلانة' a Muslim name? It has origin in Hebrew. Is it proper for a Muslim? It's meaning is to soften, to make more gentle, to make more lenient.
There is no such thing as a Muslim name. There are, however, two important guidelines to be used when naming people:
1- The name should not be offensive in any way.
Narrated AbudDarda': The Prophet (peace_be_upon_him) said: On the Day of Resurrection you will be called by your names and by your father's names, so give yourselves good names.
2- The name should not imply shirk, i.e. it should not imply worship of anyone other than Allah. The famous sahabi Abu Huraira was named 'Abdu-Shams' (worshiper of the sun) before Islam. When he entered Islam, the Prophet changed it to Abdul-Rahman (worshiper of the Most Merciful).
The fact that it's in a language other than Arabic is not objectionable in any way. The names Moussa (Moses), Issa (Jesus), Ibraheem (Abraham) and many others are Hebrew and are quite proper.