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. 😘