The flights are cheap in January because of the rain - it's low season, and fewer tourists want to go.
I was in Bali at the end of February last year, which is something like the second-rainiest month, and I can't recall much rain. I remember a few overcast days but overall the weather was pleasant. Of course, that's just a random snapshot - you could end up with a week of rain like your friend, or you could end up with perfectly fine weather.
It depends what kind of holiday you are after. If you are a sun-chaser, then January is probably not the best time to go to Bali. If you're more about the food, shops, sights and sounds, then January will be fine for you... just pack an umbrella!