Best time to visit Kauai

I usually go to Kaua'i every year for the Xmas/New Year's holidays, although I've been in Sept and Oct also.
The weather is pretty consistent throughout the year, but it's slightly rainier later in the year, therefore September is a bit drier than November. However more than the time of the year, it is the location that counts. On the windward side of Kaua'i (north/northeast) it is much wetter than the leeward side of the island (south/southwest). Therefore Poipu and all the way past Kekaha it will be much drier. Kakaha area and the Waimea canyon look like Northern Arizona. Princeville on the other hand, it's on the windward side, therefore subject to light short showers almost daily, no matter which of the months you choose. In any case, if you are interested in less probability of wet weather, September is better than November. It will be warmer too, although it doesn't vary much. Maybe in the mid 80's in September, but even when I go in Dec/Jan, temperatures are around 80F. Carry a light sweater for the evenings (and a pocket umbrella), especially in Princeville.
Enjoy Kaua'i. The most beautiful of the Hawaiian islands in my opinion. And don't miss to do a helicopter tour. Not cheap but worth every penny.
Currents on the north side are treacherous, therefore keep an eye on the kids when they get in the water.

Since you'll be including time on the north shore I agree with Roberto that September or maybe early October is your best bet for sunny weather. The North and East sides of all the Hawaiian islands are the rainy sides thanks to the effect of the predominant NE tradewinds, and Kauai is particularly affected by them. In Princeville you'll run the risk of several days of rain if you time your visit much after mid-October.
The South and West side are usually pretty sunny, so you should be fine in Poipu.

We’ve been to Kauai multiple times but have only been there in early April. As was mentioned above you may encounter more rain late fall. We used to stay in Kapaa. When it rained we would head to Poipu, the driest area. Princeville is beautiful. Good choice as well.
https://www.hawaii-guide.com/kauai/articles/when_to_visit_kauai

We went mid Oct. good weather. The guide on the kayak/secret falls hike said it was just before the wet season which made the hike on dry ground rather than mud. The falls were a little thin however

According to the travel section in the US News: https://travel.usnews.com/Kauai_HI/When_To_Visit/ the best time to visit Kauai is between Sep to Nov / Apr to Jun. The problem with Dec to Mar is this is when it’s most expensive to fly and rent rooms, plus, it’s the rainy season.

Only time I was on Kauai when the weather seriously interfered with vacation plans was in late November -- but that was a major hurricane (Eva in 1982) and even then after the hurricane passed the skies cleared. And eventually the power came back on. Other trips we just took shelter when the rain started and went on with our day when it stopped a few minutes later.

I have been to Kauai in all seasons and have always had a great time. October was a nice time to go and we generally had good weather, though we did experience some rain up north. Our typical time to go is February, though it can sometimes be rainy (especially up north) and it’s a busy time to be there. I would recommend booking soon as accommodations fill up and rental cars have been been expensive and in short supply this summer. Enjoy your trip!

Personally I prefer SEP/OCT to visit. It's rare for rain to last more than a few hours, and it's fairly warm rain too. But then I live in PNW and we're used to rain.

If you can, visit Barking Sands. Fantastic beach.

We lived on Kauai for 13 years, so I have to correct the impression given by some here that the winter rains when they come are transitory and somewhat pleasant - that accurately describes summer conditions but not the winter season up on the north shore.
As Roberto mentions early on in this thread the weather can vary dramatically depending on which side of the island you happen to be on at any given moment. Pacific storms in winter can settle in for days at a time, especially up near Princeville, and Hanalei is famous for winter flooding when that happens. You can usually find the sun by driving west towards Poipu and Kekaha, but that's not necessarily a guarantee either if your visit happens to coincide with an unlucky stretch of unsettled winter weather.
Booking a visit to Princeville after around the middle of October would be a roll of the dice. September is a much safer bet.
Also, the beach at Barking Sands is terrific but you need to have a military ID to access the base. The state beach at Polihale just to the north is picture postcard beautiful but you need to check locally regarding the condition of the access road getting into it - it's an old cane haul road whose maintenance and grading by the state are sometimes sporadic at best, and your rental car contract may be void if you venture up there.

Just returned from 2 weeks in Kauai (Poipu) yesterday. Our 7th visit to Kauai. Concur with all the Sept/early Oct recommendations. Oct is fine for Poipu but iffy for Princeville. Be aware that the landslide and road blockage between Princeville and Hanalei create considerable traffic congestion around Princeville. Last Thursday afternoon we could not get to Princeville due to the long backup on Highway 560. Only one lane is open past Princeville, access hours are limited so they can work on it and weekday backups are huge. Work is expected to continue for several months. Google it for more info.

Given that access to Hanalei and Kee beach will be very restricted (parking reservations now required for Kee) I would suggest you reconsider Princeville and spend more or all of your time in the south. Poipu beaches are much better/safer for kids and access is easier.

Also be aware that restaurants on Kauai are still struggling to get back to full service. Covid restrictions should be lifted by the time you go, hopefully. Currently masks are requited indoors and restaurants are limited to 75%. But due to staffing problems hours are often limited and menus are reduced. Reservations are pretty much mandatory at the nice places - often booked out 2 months (Tidepools and Beach House, for example). Bubba's hamburger place is closed two days a week due to "staffing shortages". Quite a few of our past favorites are just gone.

Not to say we did not have a wonderful time. Just be aware of the current limitations. we are not back to "normal".

Update on road to Hanalei - local news reports say that they road will be open to two way traffic around the end of September. There still may be brief closures for construction equipment, etc.

You didn't mention the age of your children or their anticipated school situation. I did want to mention that for older children (high school aged) in a traditional school setting, that this much time off of school would really impact both their learning and their grades. That might not be an issue for your situation, but the teacher in me was concerned. :)

I'm not sure planning this late will be successful as finding an affordable rental car at this late date will be impossible and in Kauai you will need a car. I wouldn't split your time .... it's a small island.

Thanks everyone for your suggestions. My kids are 4 and 8 and with the delta variant showing up we have decided its not safe for the kids to travel without being vaccinated. We will wait for them to get their vaccines and plan for next September.

.......and just on NBC national news tonight, the Governor of Hawaii is asking tourists to stay away to keep covid cases from surging.

Here is the CNN version of the story:
http://www.cnn.com/travel/article/hawaii-covid-tourism/index.html

So, maybe next year will be the best time to go, if more get vaccinated and normal life returns...maybe.