The word googly comes from cricket and it refers to a deceptive way of bowling. The ball looks as if it is going to bounce one way but goes in the other direction. https://en.wikipedia.org/wiki/Googly It is hard to know exactly what is intended here without more context.

You think you're running things, do you? Don't stroke my mousy hair. - What does "mouse hair" mean? it may be mousy hair referring to the colour/color which is typically brown and plain - Yeah, I think what he means is don't jeopardize my deal, but I admit that one's a bit of a google. yeah that one is a bit deceptive. if you gave us the full context we could answer in a better fashion. MOUSY HAIR: adjective [usu ADJ n] Mousy hair is a dull, light brown color. He was aged between 25 and 30, with a medium build and collar-length mousy hair. Synonyms: brownish, plain, dull, drab More Synonyms of mousy.

Simon's answer is correct. BUT it's not a common phrase in British English. A native-speaker could understand it from the context, but we wouldn't necessarily use it in this way.

I am familiar with the word "googly," but not the way it is used in this context. "A bit of a googly" is not something an American would say. It sounds like something an Englishman would say. Perhaps someone from the UK can explain it.