To ponder, consider, or speculate about something. A noun or pronoun can be used between "noodle" and "over." ("Noodle" here is slang for head or brain.) As my mind wandered into daydream, I began noodling over the possibility of giving up my job and moving to Japan. I wanted a day or two to noodle over the deal before I agreed to anything.
noodle over something
Inf. to think about something. Let's noodle over this problem for a bit and discuss it at our next meeting.