Shortwave Radio: assuming your island now has power or batteries that can run a shortwave radio this would be your best option for communication.
Communications can be split into two general areas, data and voice.
Tying into the worldwide telephone network is not difficult even if out at sea. There are a number of satellite networks designed for just this purpose. However due to the geosynchronous nature of most of the satellites used they have a problem covering the poles. The main exception to this is the iridium system which uses many polar orbiting satellites to provide full coverage of the world. However if we have a data connection using satellite then it is not necessary to use voice channels over the satellite as it can all be provided via VoIP. If we are close enough to land we may even be able to lay fiber optic cables out to the island, thus providing much more bandwidth than any satellite ever could for a much lower cost.
What has just been said about voice communication can equally be said about data networks.
As voice communications can be digitized easily and carried over a data network in makes sense only running one type of communication network throughout. There are many options for this, the cheapest solution would be to leverage standard commercial hardware for the purpose. This boils down to 3 media types and a handful of standards. The media types are radio, fiber and cable. Each of these has advantages and disadvantages. Radio is very cheap to install but has problems with interference, range and lack of bandwidth. Cable, meaning Gigabit Ethernet in this case also has a range problem unless routers are used but has quite a lot of bandwidth and more can be added by just running an extra cable. Fiber is the most expensive to install but has fewer limits of range (few km for multi mode fiber and around 80km for single mode fiber) and has a lot of scope for upgrading bandwidth (WDM can be used to send multiple signals down a single pair).