My neighbor childhood friend was georgian and called me this way but these days I didn't pay attention to the meaning as it looked very friendly. Now, a close friend who temporarily lives in Georgia just named me the same in her SMS - so I had motovation for search and discovery.
"Genacvale marks the process of adopting a person's sorrow and is translated as 'I die for you'. Genacvalos deda means 'I take mother's place'. Boeder (1988) offers his definition as being 'I take your place'. Genacvale belongs to the special formulas of sympathy which describe a strong emotional content of self-sacrifice: First, there are the abundant, often used formulas whose semantic fundamental pattern states, at least etymologically, the fundamental pattern states, at least etymologically, the following: the speaker wishes to adopt the negative (the illness, the misfortune... ) from which the addressed person is suffering. The addressed person's misfortune should be conveyed to the speaker; he/she wants to carry his/her sorrow for the suffering person representatively. (Boeder 1988: 12; translated by Helga Kotthoff) Boeder (1988) ascribes such formulas of self sacrifice to the entire Near East. In this context it can mean that the mother suffers for and through her child and that the singer will take her place in suffering. In certain contexts genacvale can also be interpreted simply as 'darling'. Lines in which the singer only cries Marina, Svilo, Marina recur in the lament. Repetitions facilitate the memory for the speaker and her listeners. Supports of this kind are typical for the oral discourse rather than for the written. "
Source "Language and gender research : review, reflection and new directions - Volume 3, Issue 2 - Page 53"
What an ancient and emotionally rich language and honorable culture!