I'm not sure if it is a nuance of the word 「ごねる」 which means to make difficulties or to grumble.
Here is the sentence containing the word.
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ごね出した【だした】 means "started to grumble." 出す is one of the syntactic compound verb elements. After the masu-stem of a verb, it means "to begin to ～", similarly to ～始める. See: What is the difference between 出す and 始める when used as a suffix?