Why is the bride's name bleeped in Kill Bill Vol 1?

Well for the first film and for most of the second, the Bride is on a revenge mission where she is hunting her victims. Her identity would have to be a secret to make sure she doesn't get followed or caught.

It could be Tarantino's way of breaking the 4th wall and including us in the element of mystery and disguise that the Bride has to undertake to remain anonymous. Also, this could mean we are meant to detached emotionally from her as a character until her name is finally revealed as Beatrix Kiddo by Elle Driver.

This is when we see the character as fully vulnerable and when she opens up her emotions. So Tarantino thought of this as the opportune moment to unveil all secrets of this character.

Or perhaps it was just Tarantino being weird and awesome.

From the script:

From an interview with Uma Thurman:

Why do they bleep your name?

That one eludes me. You'll find out her name. You will definitely find out her name, I can tell you right now, but that'd ruin it.

From an interview with Vivica A. Fox:

What name do you and Uma say when they bleep it out?

Beatrix. Her name is Beatrix. It is weird right? Even when I saw the screening of it the other day I thought, “Are people going to get that?” You never hear her name. She’s just The Bride. Whenever I go, “Sweetie this is an old friend of mommy’s, Beatrix, Beatrix, this is my daughter,” it’s bleeps. And she goes, “Hi honey, my name is…” and it bleeps. See, you’ve got to go see it again.

Why does he do that?

Because he wants you to come back and see it again and figure out what the hell her name is. Just to add little things. Just a quirk.

There were several hints about the name, giving people the opportunity to figure it out before it was eventually revealed in Volume 2.

For example, when she buys a plane ticket to Japan her name is visible on the boarding pass:

Where did the name Beatrix Kiddo come from?

From an interview with Quentin Tarantino:

Uma came up with the name Beatrix -- she worked for somebody with that name. And I came up with Kiddo. That's what I call women -- when I really like a girl, I call her "kiddo".

Speculation:

Quentin Tarantino is a known fan of Jean-Luc Godard (his production company A Band Apart was named after Godard's movie Bande à part).

In Godard's movie Made in the U.S.A. there is a running gag:

Godard censors Richard’s second name with all kinds of sounds possible.

So maybe this was Tarantino's inspiration.

Her name is bleeped because she cannot be named until she deserves to. Throughout the film she has 4 names. In chronological order the first event of the story is the wedding massacre. At this point she is Black Mamba. She is shot in the head by Bill. This is the death of Black Mamba. She wakes up four years later. This is the birth of The Bride. Now The Brides sole purpose is to take up the sword and wreak havoc on those who wronged her. She kills O-Ren with the sword and Vernita with a knife.

She becomes dependent on the sword. So much so that she underestimates Budd. Her plan is of course to attack Budd in his trailer with the sword. This immediately backfires when Budd shoots her with rock salt and disarms her. What happens next is the most significant part of the film. Having her sword taken away from her she is now powerless. Budd nails her in a wooden coffin and buries her alive. This is the death of The Bride. It is during this burial that through remembering her training with Pai Mei she realizes she doesn't need to rely on the sword. Her power lies in her hands.

She punches her way out of the coffin. This is the birth of Beatrix Kiddo. From here on her name will not be bleeped and she will not use the sword again. Beatrix doesn't kill Budd, Elle Driver does. Beatrix snatches Elle's other eye out leaving her completely blind. It is questioned in the end credits whether Elle dies or not as she is left blind in Budd's trailer with a deadly Black Mamba snake. After this, only Bill is left. She tracks him down ready to kill him when suddenly she is killed. Well symbolically of course. She is shocked at the sudden realization that her daughter is still alive. Her daughter shoots her with a toy gun. Beatrix falls to the ground (dies) and is woken up by B.B. (her daughter).

She is reborn as "Mommy".

An explanation based on secrecy from other characters isn't self-consistent or internally consistent. She is hunting people who know who she is and beeping the name for the audience doesn't stop the characters knowing who she is (the name isn't beeped for the characters but for the audience). And the people she is hunting know they are being hunted by her.

Careful attention to what characters say to her before her name is revealed to the audience, however, suggests a different explanation. Several times she is referred to as "kiddo" by other characters, especially Bill, which the audience will interpret as a term of endearment if they don't know her name is Kiddo. They can then reinterpret the earlier scenes when the real name is revealed.

This subtle misdirection seems a plausible motivation for Tarantino.

I thought it was a humorously heavy-handed ploy to evoke Clint Eastwood's "Man with No Name" character in Sergio Leone's Spaghetti Western films.

It's basically just a reference to 'Made In U.S.A' and...that's pretty much it. That's the only reason anything happens in that movie; it's just there to remind you of another movie.