How to connect RFduino to internet?
It sort of makes sense.

There isn't much of a problem keeping the Uno on 24/7 (it doesn't consume much energy). Instead of connecting it to a battery, get a good wall wart for it. This way, you don't have to worry much about synchronizing the RFduinos as well. Just have it wait patiently for an RFduino to connect to it. However, if you do want to keep them in sync, then just use the Time library, somehow turn them all on at the same time (easy if you use the reset pin), and pray that their power source is uninterrupted.

About the RFduinos: you probably can keep them on as well, on a loop that breaks every 10 minutes to turn on bluetooth and send data. Even better, use a library like this one which gives you a power efficient sleep mode to work with.

If you don't want to keep the RFduinos on, though, a simple circuit using a 555 timer and a relay ought to suffice to turn it on every few minutes (it can turn itself off via the RESET pin of the 555). As long as the Uno is on 24/7, you need not use any special sync and can have the RFduinos sending values at staggered intervals.

Here's a schematic:



Connect the part that says "To Pin" to some pin on the RFduino. Send a constant HIGH output on it. When you want to shut down, send a LOW signal.

The "To Vin" and "To GND" parts go to the respective pins on the RFduino.

If you can't find a 866K resistor, something of a nearby value ought to work. Basically, ln(2)*R2*C2 should be 600 seconds.

Note that 555s have different RESET behaviors. We want the one which goes on it's LOW cycle when reset.