The Super Saiyan Dragon Ball Z series began as Manga created by Akira Toriyama. A sequel to Dragon Ball Z and Toei's Dragon Ball GT. In the Manga series both Trunks and his mom, Bulma had the same purple hair color. But when Toei, bought it they decided to make some funky changes. To liven the show a bit, changed mom's hair blue but made no changes to Trunks.
Why? No one knows for sure but there's huge speculation about the reason for this. Many believe it to be a fashion statement. Blue hair was in. Especially for the ladies. This was a chance to build a connection with the female audience. Or, The Toei studio couldn't tell from the comic which color to choose. So out of compromise used both colors, blue for mom and purple for the son. In either case, the choice gave room to speculate among fans. Later on down the line.
To maintain consistency with family lineage, Toriyama changed Trunk's hair blue. Since then, there's been a back and forth between Toriyama and Toei about hair color. A compromise was to work it into the time travel narrative. Present Trunks maintains his purple locks while Future trunks sport blue hair. But how does this work for the story?
Blue: The Fan Fiction
Time travel is a major part of the DBS storyline. Since Future Trunks, is a time traveler, time travel caused his hair color to fluctuate. This is laughable but hey, it's fiction, anything can be written in.
The actual storyline kind of goes the same way. Future Trunks went back in time so much, he created alternate timelines. In one timeline, he's happy and everything is all good. In the other timeline, he suffers something major. To combat his suffering, Trunks yet again travels through time. He's worse than the Flash trying to save his mother. Trunk's color flux may have nothing to do with the storyline at all. Contractual issues could weigh in here. Toei couldn’t make changes to the Trunks character but had room to adapt otherwise.
Another common consensus among fans: hair color changes from childhood to adulthood. Ha! Too funny. It's rare for hair color to be bipolar. But again, we could go back and forth on this all day.
Blue: the Non-Fiction
So the real-world reasons? One, creative vision. Every artist expresses their passion differently. These differences will show in the finished work. Also Adaptations! Like fanfiction, an adaptation shows an alternate view of the storyline. It also gives creators a way to explore character concepts to draw in new audiences. Cater to different age groups and socioeconomic backgrounds. Expansion needs writers, each with their own ideas. The Dragon Ball Universe is over 30 years old. Not, the oldest comic but old enough to change hands quite a bit. These factors play into the Trunks we see today.
Two, money. There's a huge profit in adaptations. Ask Netflix how much they're funneling in from them. Let's not forget about the toys. One of the most profitable businesses in the history of entertainment is the toy lines. Disney makes millions from their entertainment franchises in toy sales alone. Toy manufacturers like Funko rake in similar profits from DBZU collectibles and wears. Selling Trunks with blue and purple hair has its advantages. Dragon Ball fans love the changes. Change fuels creativity, year-long rants, and fresh content. There's always something to say about Dragon Ball.