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Assuming Diminick was anglisized from Croatian: I went through the Croatian Wikipedia and found Diminići, the name of a city, so maybe it is a toponymic surname. Potentially, if it was founded during the Communist-era, the name of the city could have come from the name of a hypothetical high-ranking party member, Diminić. I can tell you that Dim most likely comes from smoke. I suspected that, as the Polish cognate is dym. Maybe "Dimin" (smokey) was either a nickname or given name of your Croatian paternal ancestor. As opposed to Western and Eastern Slavs, Southern Slavs are keen on single-lexeme names. Mladen (Young) and Dushan (Soul) come to mind.

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https://hr.wikipedia.org/wiki/Diminići
https://en.wikipedia.org/wiki/Slavic_names