My family is of Sicilian heritage on my mother's side. As a child my mother and grandmother would pinch my cheeks and say in a pleasant high pitched voice "bedda nica ducci." I know it was a pleasantry, but don't know what it means except the bedda part. I'm also not sure of the spelling. Anyone also familiar with this phrase?
If you think the spelling "bedda nica duci" could be correct, I would translate that to : "Pretty, sweet toddler/young child".
Nica is a young female child. I think the word can be used both as a noun and an adjective, though the noun is often changed to "nicaredda" to make it even more endearing. When I used to hear it in that context, I understood it as a noun. Baby is "picciruddu", toddler/young child "nicu", older child "carusu", adolescent/young adult "piciottu".
If you feel the double cc consonant was very pronounced, then it might be possible it could have been misunderstood and would then be a diminutive or a regional variant that I've never heard of. Diminutives I usually heard were "etta" or "edda" but never "ucci" - I have never heard or seen that. I have heard "uzzu". They usually depend on the preceding vowel sound and will be adjusted to end in -a if feminine and -u if masculine.
The diminutive for nica is nicaredda.
From you discription, I understand your family members were pinching your cheeks saying the equivalent of "what a cutie-pie".