offer for vs. offer to

Both are OK, although "of" would be better than "for" in example 1.

The confusion arises because the word "help" can be either a noun or a verb.

In the first, "help" is a noun. A close paraphrase would be:

He refused the organisation's offer for assistance.

But the following sounds better and more natural:

He refused the organisation's offer of assistance/help.

In the second, "help" is a verb. A close paraphrase would be:

He refused the organisation's offer to assist.

The word “for” is usually used to show:

• Purpose e.i. “I wear old trousers for painting.”
• Denote a receiver of something e.i. “She bought a teapot for her sister.”
• Express duration e.i. “I usually go there for a couple of hours.”
• Indicate exchange e.i. “The shirt was sold for a dollar.”
• Replace “because” e.i. “I missed school yesterday for I was sick.”

Since Number 1 does not fall under any of these uses, therefore, the correct answer is Number 2.

Context is important here.

He refused the organisation’s offer for help.

The organisation requires his help. They are offering him something in exchange for the help. It is not acceptable. He refused the organisation’s offer for help.

He refused the organisation’s offer to help

The organisation wants to help him and he doesn’t want it.