The name of the show is obviously iconic, but always felt a little mismatched with the first part of the series before Saul is introduced in full.

Its funny seeing people who never saw BB questioning why the show is named Better Call Saul when in the early seasons, there is nobody calling Saul, let alone the existence of a Saul, for so long. To me it went hand-in-hand with the well-documented writing process of the show, which was originally intended to transition to Saul quickly and center on his office hijinks.

But then that didn't happen, and the show became much more about Jimmy, but the name was still Better Call Saul. Part of me even thinks they added the bit about Francesca needing to be at a payphone in the "Quite a Ride" fast-forward intro as a way to better tie the "Better Call Saul" motif into the DNA of the show, since up to that point it didn't seem like the most appropriate name.

But now, I think the title is perfect. The entire show is about Jimmy building this Saul persona as his defense mechanism, a place to hide from pain and guilt and shame and inferiority. "Better Call Saul" isn't so much referring to all the sleazy clients or the eventual classic commercial slogan, its referring to Jimmy. Jimmy summons Saul. Jimmy calls upon this manic part of himself to allow him to cope with his reality. One way of looking at it is that Jimmy is actually Saul's "first client". The same way Jimmy superstitiously uses his jazz hands routine to conjure up phone calls on his nail salon phone when he is hitting a rough patch, he conjures up Saul as well.

The scene with Jimmy's "LIGHTNING BOLTS SHOOT FROM MY FINGERTIPS" tirade is the best encapsulation of this idea. Jimmy is backed into a corner and instead of reflecting and taking a deep look inside himself, he thinks it better to call upon Saul, and his tantrum ensues.

Heartbroken from Kim leaving and can't bear to show it when she comes to sign the divorce papers? Better call Saul and act like you don't care.

A goofy cab driver living with his mom is trying to pull a fast one on you? Better call Saul and outcon him before he can con you.

Stuck in a holding cell with limited options and no way out? Better call Saul and remember who the hell you are. You're SAUL GOODMAN for god's sake!

TL:DR -- Better Call Saul is a much better title for the show when you think of it as Jimmy "calling upon" his Saul persona like he is summoning some kind of mythical figure, as opposed to just a cute reference to the BB commercials and the iconic slogan, though that is obviously also part of the meaning.