>>>I am actually trying to build an argument to persuade a friend he should only switch it off when going to bed
Well there's a very good argument as to why he should NEVER turn it off:
BT OpenReach's technology monitors connection speeds, to determine what speed the line is capable of handling (and then delivering that speed). When you first connect a router you probably won't get the maximum speed for the line; it gradually rises (as the system recognises that the line can handle it) over a period of about 10 days.
If a router is disconnected, the system sees it as a fault on the line and reduces the speed supplied to the line, back to the 'starting point' level, when the router is then reconnected.
So, in order to ensure that you maintain the maximum download speed for your line, you should NEVER disconnect a router!