Is hooking up led lights to a timer cost effective?
No, the lights will use many more times the power that a timer, digital or analog, uses. If you talk about digital timers, the difference is astronomical. A modern solid-state timer will use no more than half a watt (5v logic x 100ma consumption), probably a lot less.

In terms of mechanical timers, it's still a lot lower than 7w; Using a kill-a-watt, I just now plugged in all 4 mechanical timers I had around (at once), and the meter was still unable to measure the draw, placing all 4 at under 3 watts combined. There are different brands in that batch as well: 1 intermatic, 2 ikea, and one unknown. Think how long a tiny battery can power a mechanical watch; clocks are low-power these days.

Obviously 7w > (x/&lt;3), so your friend is simply wrong. He's a lot closer than he would have been 5 years ago, but he's still wrong.