Methylene blue is a very effective water-soluble blue dye and you can buy it on amazon quite cheaply (the pack at that link is probably enough to dye a table and chairs). That might do the trick. It's also used for treating fish in tanks, so is quite benign. The question becomes one of whether the water raises the grain too much given that the dyed layer is likely to be quite thin. It's also soluble in alcohols (including isopropanol, so surgical spirit/rubbing alcohol) which wouldn't raise the grain.
I've done a quick test on a bit of scrap softwood using something like a 0.7 mmol/l aqueous solution (0.2 grams per litre of water). It was brief enough to not raise the grain even where I left it on the longest:
I've adjusted the colours to look about right on my screen. The wood was quite old and yellowed. It had just about dried when I took the picture.
The left for 1 minute test was meant to simulate dip-coating. I didn't hold it very steadily, so the unevenness below the "p" in the annotation was casued by the solution running.
It could work nicely, but you would want to make sure the wood is very white to start with, and stays that way, to keep it really blue instead of turning greenish. This is probably true for any blue dye. This article on bleaching wood implies that you can bleach the original wood with the right wood bleach, then neutralise and rinse and there shouldn't be any bleach left to react. Here's some more discussion on bleaching and coating so it doesn't turn yellow; in summary: use a water-based varnish over the top so the varnish also doesn't turn yellow, though pine will yellow slightly whatever you do.
Of course, test as much as you can on scrap (and the same type of wood) first, in particular if you decide to bleach then dye.