Does 有了 have a slang meaning of "I know" or "I have an idea"?
Google translator and archchinese states the word 有了 means 'with'. However when I was watching 大耳朵图图 there was a part where 图图妈妈 said the word 有了 and a lightbulb popped up over head. Based on context it doesn't make sense that she was blurting out the word 'with' so I am guessing the word 有了 has an alternate meaning of "I know'" or "I have an idea". Is this correct?

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You guess it right. "Eureka!" is the word you are looking for.

Another similar but more in written form goes to "頓悟".

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对的~
“有了”在这里指Something occurred to me~
你在看大耳朵图图吗？很有趣。

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有了 literally means "have it/ got it". Well yes, not really a slang (well, sort of). In that situation, its a shortcut to say "I got it" meaning he/she got an idea.

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我想在这里“有了”应该是表示“我有主意了”或者“我有办法了”，which means "I have an idea"。这不算是slang，只是口语化的表达