two (maybe 3) locations would be better than 5; adding a couple of nights near Hilo will get you the VNP, Hilo, day trips along the Hamuaka coast (plus other scenic drives, and south to Kapoho, Lava Tree State Park, etc. Nice shoreline access there, also.
THen from Kona you could go south to South Point, etc. and visit the Place of Refuge, Captain Cook monument (nice boat excursion to snorkel), coffee plantations, whatever. If you want a third spot, go north to the beaches of north Kohala.
Relaxing to me would not be uprooting more than once!

So, Kona and/or Kohala coast (Waikoloa, then the beaches of the coast) would be my first choice.

We have gone to the Big Island for many years--usually for 2-3 weeks. Our schedule is to spend 3-4 days in Hamakua--usually at a b and b in in Honoka'a. That gives us easy access to Waipio Valley, north Kohala and the charming town of Hawi, to Waimea for our yearly dinner at Merriman's and if we want a beach--Hapuna is only about 40 minutes away.
Then 4 or so days in Puna--usually at Kapoho--we stay at Lagoon House www.lagoonhouse.com --but doubt it would be available this close to your time there. (But there are many other rentals available in the area) This give us easy access to both Hilo and VNP and to Paolo's restaurant in Pahoa for a lovely Italian dinner prepared by a chef from Tuscany.
We then drive the southern route to south Kona (past Punalu'u's black sand beach, lunch at Hana Hou cafe in Na'alehu), where we spend the rest of our time on island. We have some favorite rentals in that area with views of Kealekaua Bay and easy access to some great snorkeling areas, to Kailua town, and even the Kohala beaches are not prohibitively far away.

Yes, two locations will be enough. You'll be with friends Hilo-side, then move to the Kohala area. (We've also stayed Keauhou, but Kohala is more likely to be free of any vog, and there are *beaches*.)

We've never stayed on any of the four major islands for less than two weeks; often stayed four.

Reserve your rental car today. *I* would plan to fly into ITO and out of KOA.