I met with a structure of this type many times during last days, so I wonder:
What is the meaning of "[verb] 一 [verb]"?
It means "for a moment", in other words, doing the action for a short time only.
讓我看一看 ==> Let me see(for a moment). 你可以想一想 ==> You may think(for a moment).
It is often used as suggestion to the speaker with mild imperative.
看一看吧！ ==> Just look(for a moment)!
Duplication of words is very common in Chinese. It doesn't have much grammatical meaning but it adds a little emphasis.
Sometimes, especially for Chinese spoken in Singapore (my native tongue), the order [verb][verb][particle] is used more often for better cohesion.
For example: 让我想想一下吧! （Usually 让我想一想吧!）--> Please let me think for a moment. 你的伤这么严重，让我看看！(Usually 你的伤这么严重，让我看一看!) --> Your injury is very serious, let me see.