Actually, the name 干妈 means 'godmother', not 'grandmother' or 'honorable mother' as claimed upthread.
Literally 老干妈 (lǎo gān mā) translates as 'old dry mother', here meaning 'dry' as in the sense of not offering liquid sustenance to the child. A bit akin to being the opposite of a wet nurse, I suppose.
However, unlike 'godmother' it carries no religious significance or responsibilities.