Hello and welcome
As you probably know, most accommodation on Symi tends to be rather small so catering for 8-10 people really cuts down your options. Symi Visitor Accommodation has some large houses to rent which may accommodate your party but they are unlikely to be interested in such short-term lets since it isn't financially viable to rent a house for just a couple of days given the fixed cleaning and transfer costs. But it's still probably worth contacting them at email@example.com
to see what they suggest. Since you're looking to visit 6-8 times you may be able to come to some sort of arrangement with them.
"Different" or "unique" hotel accommodation is possible but won't to come cheap! Somewhere like the Aliki or the Old Markets are options that fit the bill but you will pay a lot for their special charms.
For more "normal" hotel accommodation you could look at Nireus or the Opera House in Yialos. Perhaps, if you don't mind modern, Iapetos would be a possibility too.
In Chorio, the Village Hotel and Fiona are small places but they should be able to cope with your party. Down in Pedi you have the Pedi Beach Hotel which is big enough to cope but it is used by package tour companies so may not have availability for all of your dates.
To be honest, that's about all I can think of. Others may chip in with more suggestions.