Blue is one of the many colors portrayed in the series. Most notably by the boy who started it all, Valt Aoi. Other characters that have blue as their main color include:

Lui 

Naoki

Ukyo

Naru

Dr. Evel

Drum

and Hikaru. (there're maybe more that I missed cuz I can't remember all of them so plz comment :))

Now let's get into the meaning of the color.

Blue often symbolizes security, trust, loyalty, responsibility and intelligence. Blue also represents confidence, peace, honesty and reliability. 

Valt, as we all know, is trustworthy and very loyal to his friends and his bey, Valkyrie (who also displays her loyalty to her blader). Heck, if he wasn't loyal, he wouldn't have saved Shu in the first place. Valt also shows a great amount of confidence in every battle. 

The color blue has it's effect on the mind. Whenever someone sees blue, it gives them a sense of protection, calmness, relaxation and support. So if ya feeling blue, Valt Aoi is there for you! (kinda ironic). Ukyo kinda represents this too since he has a calm demeanor.

Naoki and Dr. Evel's color scheme of blue fits well with their jobs. As said before, blue can symbolize intelligence, and since both of them are scientists that research beyblades, they gotta be really smart. Same goes for Hikaru except he isn't a scientist and mostly comes up with strategies to win a battle. (Ravenclaw *cough cough*)

Naru Akaba is portrayed with blue in her color scheme as she is very supportive of her older brother, Aiga when he created Z Achilles aside from contrasting Aiga's red theme.

Now let's go to our favorite shark teeth boi, Lui Shirosagi. Aside from sometimes being calm, confident and intelligent, Lui's personality doesn't really fit most of blue's symbolism. Despite having blue as his main color, Lui has an intimidating aura, not carrying the psychological effects that blue usually has in contrast to Valt.

So why is he dawned in blue even though he's intimidating?

Well, we need to look at his hair.

Lui's hair looks like fire. In nature, blue flames burn hotter than normal orange flames. So if we take this into account, Lui's blue represents his hot-headedness (pun totally not intended) and his desire to be the very top of the blading world. Drum shows this burning passion too since his resonance aura is blue.

Speaking of blue flames, let's go back to Hikaru. Aside from representing his intelligence, Hikaru's blue may symbolize his position as the older brother of Hyuga, since again, blue fire burns hotter than orange/red fire. This blue fire, red fire concept may also apply to their beys, Helios and Hyperion.

In the anime, Helios is mainly blue in contrast to his father, Hyperion's red. In mythology, Helios is the god of the Sun and is usually depicted with gold attributes like a chariot and a radiant crown. In real life, the Sun appears yellow to our eyes. Why is Helios colored blue despite the fact that he is mostly depicted gold? Aside from matching with his blader's main color, perhaps Helios felt overshadowed by his father's light and wanted to be greater than him, thus he burns hotter with blue flames. His theme with blue and fire can possibly get us an idea of his true personality (usually bey spirits appear all wise and god-like but I like to think that they have personalities of their own unbeknownst to their bladers), calm and reserved at most times but gets pissed if someone is able to push his buttons (I'm lookin at u Lane).

So, that does it for significance of blue in the series. Some of these are just theories so the symbolism is up for interpretation. I probably got some stuff wrong (I'm not a professional, ok? Just doing this to free my imaginations) so feel free to correct me in the comments :)

Next color: Red.