The only one that is correct (grammatically) is the last one but it sounds slightly odd to offer a rest. You might offer someone an opportunity to rest, a room to rest in, a bed to rest on, but the rest itself doesn't quite work in this sentence. You don't quite seem to have got to grips with the meaning of 'offer'.
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Nona The BritClare offered to take a voyage to him before the voyage
You can offer to do something. You can offer a thing or a service. But you can't offer that someone else do something.