Is there a word for the ideas you get while showering?

try eureka moment or eureka effect - the original Eureka ("I've found it") moment is named after a myth regarding Archimedes who came upon an epiphany while being in a bath (is that close enough to your shower requirement?). Anyway, Urban Dictionary has the word showertime epiphany for exactly what you're looking for:

An epiphany, a shocking revelation that occurs when doing the most mundane of tasks, such as showering. This could be the solution to a long-unresolved problem.

try also showphiphany or shower epiphany

The word breakthrough is defined as

a sudden increase in knowledge, understanding, etc. : an important discovery that happens after trying for a long time to understand or explain something

Of course, breakthroughs occur in other circumstances besides while showering, but it might be helpful to understand the conditions that are present that often lead to breakthroughs:

The article "Why thinking in the shower may be an ideal model for 'creative pause'” discusses the term creative pause, define by Professor Lajos Székely, in his paper for International Journal of Psychoanalysis, as

the time interval which begins when the thinker interrupts conscious preoccupation with an unsolved problem, and ends when the solution to the problem unexpectedly appears in consciousness.

In that article, it summarizes several factors present while showering that promote creativity while encouraging a temporary reduction in one's attention to the problems the problems at hand.

It goes on to enumerate several factors that might encourage this process:

• There’s little opportunity for distraction.
• Minimal mental engagement is required for the the task at hand.
• Showering creates a “white noise” effect.
• A change of scenery sets the stage for the unexpected.

I'll leave it to the article to provide the details, which can be obtained at the link. (The topic can also be researched through the other references provided within this response.)

The subject can also be found having been discussed by Dr. Edward de Bono, who originated the expression lateral thinking. In his book "Serious Creativity: Using the Power of Lateral Thinking to Create New Ideas", he says

Some of the best results come when people stop to think about things that no one else has stopped to think about.

Obviously creative pause or creative disengagement doesn't only happen in a shower. It describes the conditions that lead to creative ideas; conditions where one can distance oneself from their challenges and cleanse their mind for a moment or longer and refresh their thoughts on the matters that concern them.