The phrase "use your noodle" simply means "think about it.".
To noodle (around) on something, while it does make use of the noodle (= head), may derive from the regional German nudeln, to improvise a song, or from the late-19th-century Scottish sense of noodling as humming a song to oneself. By 1937, to noodle was to fool around with notes to create music. By 1942, perhaps by association with the doodle (a loose, free-associating drawing), it acquired the broader sense of messing about with words or ideas without a clear goal.
What is it about food and the human head that makes them constant companions? The head has been called the noodle, the loaf, the bean and other food-related terms, although they didn't all begin as food references.
When the 16th-century painter Giuseppe Arcimboldo created heads from artfully assembled fruits and vegetables, he divined the truth that we are what we eat. Shakespeare's head-related insults included tripe-visaged (having a face like a cow's stomach) and beef-witted. In his book Carnal Knowledge, Charles Hodgson says even the Old English word for head, heafud, "is thought to have been pronounced something like 'hey-ya-food.'"
From 1720, a noodle was a stupid person, but nobody is sure how that idea emerged. By 1762, the noodle referred to the head itself, stupid or otherwise. Laurence Sterne used it in Tristram Shandy: "What can have got into that precious noodle of thine?"
- source: edited from https://www.theglobeandmail.com/arts/books-and-media/language-use-your-noodle-old-bean/article4381047/#
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Came across the phrase by accident. One WXC blogger-friend I knew posted a post recently with humorous sentences to represent the alphabet, but only the first 11 (from A to K) of them. I found it funny, so tried to add on under his post for letters "LMON", and then found "noodle" 
See: https://blog.wenxuecity.com/myblog/76782/202409/23106.html
Wish you all a happy Tuesday with a noodle dance! ;-)
Note “The Noodle Dance” is a song and dance routine performed by (and possibly created by) Peanut, Jelly, and Baby Butter in "PB&J Otter" to help them think of ideas when they are stuck on a problem.
