how much space are you talking about?
what is the soil like?
What is there already?
How much work do you want to put into it?

Runoff is not good for a "true" bog garden. That would be acidic low nutient, growing things like CPs and orchids.

Sounds like a nice place for a rain garden or perhaps marginals if it stays wet enough often enough.

It gets moderate morning sun and then it pretty much shady thereafter. I would say the length of the area from fence to the side walkway is about 18' to 20' and it would border an existing bed. I was just going to dig down a foot and a half and put in drainage rock, the line the "dry creek bed" with liner then rocks and pebbles.

But if putting in a bog would be easier, I am open to it. I was looking for a FAQ here, but could not find it.