Since I don't have first - hand knowledge of this fanfic, my best guess is that it could involve a situation or concept that is more 'blue' in nature compared to the TARDIS. Maybe it has to do with a different kind of blue - hued alien or a blue - tinted dimension that is explored in the story. It could also be a play on words, where 'bluer' refers to something other than the color, like a 'bluer' state of mind, and the story then weaves this concept in relation to the TARDIS, perhaps showing how this 'bluer' thing interacts with or is different from the TARDIS.
I'm not sure specifically as I haven't read this fanfic. But generally, if Harry is related to the TARDIS in a fanfic, it might involve some sort of crossover where Harry from the Harry Potter universe has an encounter or connection with the TARDIS from Doctor Who. It could be a really creative take on combining two very different fictional worlds.
Yes, there are likely some fanfictions out there with this concept. Fanfiction writers often come up with creative crossovers, and combining the magical world of Harry Potter with the time - and - space - traveling TARDIS from Doctor Who is a really interesting idea that some fans might have explored in their writing.
Obviously, Clara and the TARDIS are the main characters. Clara, with her inquisitive nature and strong will, and the TARDIS, the sentient time - space ship with its own mysteries.
A really interesting 'Doctor X Tardis' fanfic I've come across is 'Tardis Tales'. In this fanfic, the Tardis has a sort of consciousness that the Doctor tries to fully understand. It delves into the mystery of the Tardis' inner workings and how it has a connection to the Doctor on a deeper level than just being a ship. It's a short but engaging read that will make you look at the Doctor - Tardis relationship in a new light.
You could try searching on popular fanfic websites like Archive of Our Own (AO3) or FanFiction.net. These platforms usually have a large collection of fanfics, and you might be able to find this particular one by using the title or relevant tags in the search bar.
The concept of time is also treated differently in this type of fanfiction. While the Time - Turner in Harry Potter allows for a limited form of time travel, the TARDIS can travel through all of time and space. Fanfiction writers often explore how these two different time - traveling mechanisms interact. They might have the TARDIS enhance the power of the Time - Turner or use the Time - Turner to fix a problem within the TARDIS's time - space continuum. This exploration of time adds a whole new layer of complexity and uniqueness to the stories.
One interesting plotline could be Clara discovers a hidden chamber inside the TARDIS that contains secrets about the Doctor's past. Maybe it holds memories or artifacts that the Doctor has long forgotten or tried to keep hidden.
Maybe he discovers some ancient magical - technological hybrid plans. He could find them in a hidden part of the Hogwarts library or in an old wizard's lair. Then, with his knowledge of magic and his friends' help, he starts to piece together the parts, using magic to substitute for some of the more advanced technological components that are hard to come by in the wizarding world.