Store was out of all blue baking decor.
Can't think of way to dye them blue and be appetizing.
Maybe use alternate design scheme?
If they are cut-out to be 6-pointed star, just add silver or gold dragees on top of white icing.
If they not star-shaped, ice a star on top of cookie, then add glittering dragees.
I agree with Nancy about not coloring the actual cookies blue.
How about using blue sanding/sparkling sugar on top of the cookies?
How about decorating with blue and white icing?
I also wouldn't dye the dough itself. Blue is a tricky color for food. The star of David shape would be enough to channel Chanukah for me, but if you really want blue, I might go with a line of piping, with a sprinkle sanding sugar, or with gazing/dipping/flooding the tops (or a portion thereof) in blue.
If you over mix raspberries or blueberries in a dough it turns blue... But, I agree with the other comments. Even if you have blue dough- when it bakes it just looks unappealing and vile.
This site has some ideas: https://toriavey.com/toris-kitchen/hanukkah-holiday-sugar-cookies/
Frosted would probably look the best. But, if you don't want a heavy buttercream, I just made a simple milk and confectioner sugar icing (but on the thicker side), and add the flavoring off your choice. Then you can either color it a solid blue or instead of mixing in the food color, swirl the drops a bit, using a toothpick. This will give a marbling effect on the finished cookie. You can then simply dip the top of your cookie in the icing and place on a baking rack to allow the icing to dry. Very good with a sugar cookie.