I remember him saying that / say or tell

They are both correct, but they have slightly different meanings.

"him saying vs. he said" - The nuance on "saying" is the memory of him talking.  "He said" - the emphasis is that he had plans. They are interchangeable but the feeling of the sentence is so very slightly different. 

"them making" vs. "they made": This one has a more apparent difference.  The first sentence is a memory of them "making" a boat.  Maybe you saw a picture of them making the boat or you saw them make it.  "They made" - This is only remembering that they made it.  So if someone told you, "they made a boat last weekend", you wouldn't remember them making the boat, only that the boat was made.  It is action vs. result.

"Her going" vs. "she will go":  This is just past vs. future.  "her going" is in the past.  "she will go" is future.

One more thing,

I'm confused say and tell.

For example,
1)He said, "I am going to cook".
2)He said he was going to cook.

"This is a direct vs. indirect quotation." "I am going to..." are the exact words out of his mouth. It is the difference between 

a. 그는 나한테 요리하겠다고 말했어요

b. 그는 나한테 나는 요리할까라고 말했어요

I think.

There are a couple of differences between say and tell.  The main one is that we don't use an indirect object with said.

For example about tell.
It's the same.

1)He told me I am going to cook. - He told me, "I am going to cook."
2)He told me he was going to cook. - He told me he was going to cook.

Let me know if you have any questions.  Here is a more in depth explanation of "tell vs. say."

http://dictionary.cambridge.org/us/grammar/british-grammar/say-or-tell