It's the generic Turkish diminutive suffix, which has many functions. In this context, it adds affection to the noun it modifies; so for example "teyzeciğim" would mean "(My) dear aunt" or abiciğim: dear
brother/ "bro"
Therefore, "anneciğim" means "(my) dear
mother"
We'll make a phonetic "Jim" out of "cigim“ to spare you english speakers the
extra thinking. 😘