Good day and thank you for your question. We look forward to welcoming you and your granddaughter for your 'special' time together.
Yes, I would recommend renting a car. The Island is much larger than most first-time visitors realize. For exploring purposes there are taxis and tour companies but they can get expensive if touring every day.
As for child-friendly activities, there are many that open on weekends in early June. There is Mariner's Cove Boardwalk which hosts Ripley's Believe It Or Not Museum, Mariner's Cove Mini Golf, & Wax World of the Stars. Cavendish Adventure Zone is also open on weekends in early June. This area hosts the Haunted Hotel 6 & indoor Laser Tag. Through the week in early June you and your granddaughter can explore Green Gables National Historic Site, home to "Anne of Green Gables" house, walking trail, and gift shop. The beaches are also wonderful to explore in early June. There are 15 hiking & cycling trails inside the National Park of Prince Edward Island and there are many bike rental places along the shore in both Cavendish and North Rustico.
If you really prefer not to rent a car, I would suggest staying in the Cavendish area as this area will offer the vast majority of child-friendly activities. There is a tour company in Cavendish called Red Sand Tour and Taxi and this service can take you and your granddaughter to Charlottetown, the capital city, where you could enjoy the activities at Confederation Centre of the Arts and the waterfront.
The vast majority of accommodations in Cavendish central will offer you walking distance to the activities I mentioned as well as to the beaches and trails. There is also a grocery store in the central area of Cavendish that is also walking distance to all the activities I have mentioned including a bike rental place. There are cottages, motels, and Inns in this central area; as well as an abundance of restaurants. I would explore some of the accommodation options by clicking on the following link and then select Where to Stay. Key in Green Gables Shore and then Cavendish so you are sure to find a place that will limit your driving and allow for walking. I would suggest selecting an accommodation place along Route 6 in Cavendish Central (Route 6 is often also referred to as the Cavendish Road). Please note many accommodation providers don't list their availability for 2018 until mid-December to early January, so enter in today's date to ensure you receive the greatest selection of accommodation providers.
Have a wonderful time planning your special trip with your granddaughter and we shall all look forward to hosting you both in 2018.