생각할게요 << 생각하(다) + ㄹ게요
생각해볼게요 << 생각하(다) + 여 보(다) + ㄹ게요
Verb + 아/어/여 보다 : This pattern is used to:
1. Tell someone to try or attempt something and to see the result of doing so.
2. As a question this can be used to ask if someone has ever done or tried something, for example "미국에 가 봤어요?" could be translated to "Have you ever been to America?"
3. It can also be used to make a command sound less strong and demanding.
e.g.
이거 먹어 봐. Try this (food).
미국에 한번 가 봤어요. I've been to America once.
한번 마셔 보세요. Please try this one time (drink).
입어 보세요. Please try this on.
-(으)ㄹ게요. : Immediate Future
It is always first person and the action is usually beneficial to the hearer. This marker can only be used in a processive verb or verb that takes a direct object.
생각할게요(a little awkward)
I will think it over.
생각해볼게요(formal and honorific)
I will try to think it over = I will consider about it.