I have 2 strings of small led Christmas lights above my cabinets in the kitchen as a accent lighting (probably about 100 bulbs total) I have each hooked up to a lamp timer, and they turn on for about 6 hours a night. A friend at work laughed and said that it would be cheaper to leave the lights on 24 hours a day then to run the timers. I can not find a definitive answer, but it makes sense. Anyone know?
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/<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.
Stick a Kill-a-Watt on the string, but don't be surprised if both strings take 7 watts to run. Go measure it (or read it off the box). Rule of thumb is running a load 24x7 costs $1 per watt per year in electricity.
Suppose you spend $2 a year switching the lights on/off, or $7 a year running them 24x7, then switching is a net lose if switching costs more than $5 a year. So it depends on how much you're paying for timers and how long they last.