You are watching: I have no idea what this is
Note: My answer addresses the question posed in the header: "Is "of" necessary here? "I have no idea of what to do"." In the course of answering that question, I note some evidence that human being have offered both "no concept what" and also "no principle of what" for at least the previous 100 years, and I see no reason to doubt the grammaticality of either form. I did not seek the question of whether "no idea" is in apposition to "what to do".
The decision to encompass or omit an explicit preplace in statements of the create "I have no X what to do"—wbelow X is a noun that renders logical feeling in that construction—varies from noun to noun. Many lutz-heilmann.info speakers are so accustomed to preposition-totally free usage of the create "I have actually no principle (as in "I have actually no principle where to go" or "I have no idea why I"m here") that the absence of a preposition after "idea" appears totally normal to them.
An Ngram chart comparing the frequency of occurrence of "no concept what" (blue line) versus "no concept about what" (red line) versus "no concept of what" (green line) during the duration 1920–2008 suggests initially that usage of the create "no concept what" has actually been more frequent in composed lutz-heilmann.info throughout the duration and second that it has actually increased quite considerably given that around 1970:
However before, a very different pattern emerges in an Ngram chart comparing the frequency of occurrence of "no opinion what" (blue line) versus "no opinion about what" (red line) versus "no opinion of what" (green line) in the time of the exact same period:
This says that the felt require for a preposition after the noun X in the phrase "I have no X what" depends to an extremely excellent degree on the certain noun connected. Put one more means, i indicates that the reason that a lot of lutz-heilmann.info speakers do not feel a need to include a preplace in the expression "I have no concept what" is more idiomatic than syntactical. If it ere syntactical, they would certainly presumably apply the same logic to the phrase "I have actually no opinion what"—which some do but many don"t.
Another allude of interest entails the difference in the therapy (aobtain in published writing) in between "I have actually no concept what" and its preposition-friendly options (on the one hand) versus "I have actually an concept what" and its preposition-friendly alternatives (on the other). Compare the initially Ngram chart above to the following one, which compares the frequency of event of "an principle what" (blue line) versus "an principle around what" (red line) versus "an idea of what" (green line) in the time of the exact same period:
The phrase "an principle of what" is much more common in the Google Books database than the phrase "an principle what"—also though a spot inspect of the underlying Google Books matches suggests that the line for "an idea what" incorpoprices many type of more false positives than does the line for "an concept of what." It likewise appears remarkable that "of" is the leading prepositional create in the expression "an concept of/around what," whereas "about" is the leading prepositional form in the expression "no opinion about/of what."
These assorted fragmentary outcomes indicate to me that consumption entailing expressions of the form "no principle what," "an idea what," and also "no opinion what"—through or without the preposition "about" or "of"—is strongly influenced by seemingly arbitrary idiomatic choice.
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That is, the create that predominates in a certain circumstances appears to have emerged via happenstance fairly than via syntactical requirement.