I make tutus on crochet headbands all the time. Most my customers prefer them on it (& I do too, because it can fit from NB up to 3/4/5 years sometimes)
I use 2.5 inch, thats just what I had available and they seem to work good. I can fit a LOT of tulle on it (and the perk is because you can see how many holes you have - you know how many strips you need. I usually do 2 per knot.)
For the newborn ones Ive done - I did them about 6-7'' long, so I cut my strips double that length.
Heres some of mine on the headbands