I'm wondering in particular about contact with the Haida, since "gwai'i" (as in Haida Gwai'i, the islands of the Haida) seems superficially similar to "hawai'i" and could have been a borrowing. I'm well aware that these sorts of similarities are often coincidences, but that was the line of thought leading to the question.
Thor Heyerdahl (of Kon-Tiki fame) put forward a notion that Hawaii was colonized from the North, and that a 'substrate' of Native Americans underlay much Polynesian culture and ethnic identity (throughout Polynesia). He published peer-reviewed work on the matter in the 40s ("Did Polynesian Culture Originate in America?" International Science 1 (May 1941) 15-26). I don't think this notion has gotten much credence in the anthropological literature (which might be more appropriate than historical literature, given that any contacts of interest would have taken place between pre-literate cultures from N. America and Polynesia).
There are some interesting sources online that discuss the topic (including a page with N.A. mythology interpreted in such a way as to support the interaction between Polynesian and PNW tribes). I can tell you that among tribes I've worked with (I used to teach American Indian Studies, and have worked as a volunteer for years with various tribes on social justice and education issues) from the PNW, the link between Polynesian and N. American tribes is assumed to be correct. In particular, I've had Coast Salish elders and Haida tell me that the Maori people are considered relatives. Is there literal truth about origins there? Hard to say.
I know this isn't a science forum, but there are genetic studies in the peer-reviewed literature that link some Polynesian genes (HLA antigen regions) with the Haida/Tlingit genetic structures. You can read about it in this PLOS Genetics paper from 2008 (and in some of the references). This evidence has sometimes been interpreted as meaning that Taiwanese and Ainu people migrated to N. America, from there to Hawaii, and from there to the rest of Polynesia. With liberal genetic admixture along the way.
As a side note: I'm not sure Heyerdahl's article from 1941 exists online, but there are responses to it: a criticism of the notion from 1950; a conference paper with discussion of Polynesian aboriginal Taiwanese roots and Heyerdahl's ideas; and a recent defense of Heyerdahl.
Direct contact or direct relationship between Hawai'i and North American indigenous people is very unlikely to have occurred. However, there is solid and increasingly abundant evidence that the Polynesians contacted South America, and that cultural and genetic exchange occurred as a result of this. We are fairly certain that the Polynesians picked up some of their iconic root vegetables (sweet potato) from South America before distributing to the rest of the Polynesian network. The appearance of chickens in South America looked promising for a while as it might have indicated Polynesian contact (they brought chickens with them), but the issue is now confused because they might have been very early Spanish chickens instead. Additionally, late last year genetic findings were published indicating the South American heritage of early Easter Island inhabitants.
Overall, I doubt there was any direct contact between Hawai'i and the Americas, but there very well may have been indirect influence if the people who settled Hawai'i had previously contacted South America and passed on the results of that contact, or perhaps if ongoing contact and accompanying domestic species, etc were passed on to Hawai'i by other voyaging Polynesians after its settlement.
There is some very weak evidence for pre-Columbian contact between Hawaii and southern California. It comes in the form of the boats built by the Chumash and Tongva. Both constructed sewn-plank boats known as tomolo to the Chumash and ti'at to the Tongva. Some attempts have been made to relate these words to proto-Central Eastern Polynesian roots, the first being derived from a word for the name of tree which provides wood for canoes and the second from the word for "to sew." Sewn-boats are fairly common in the Pacific, but rare in the Americas. The only other place in the Americas that was using them at the time of European contact was along the coast of Chile, which had sewn-boats known as dalcas.
If you have jstor access and feel like reading the back-and-forth on the topic:
Personally, I side with Anderson and Arnold here. While the construction methods are similar, the actual designs of the sewn-boats used in the three locations (the Pacific Islands, California, and Chile) are actually quite different, and it seems quite strange that the Chumash and Tongva would adopt Polynesian words tangentially related to the process of making sewn-boats rather than the adopting the Polynesian word for the boat itself.