I was going through Wikipedia and found it:

Switzerland: From the toponym Schwyz (see there) first attested AD 972 as Suittes derived from an allemanic proper name Suito.
Source

Is that true?
First someone taught me that it's true. Schweiz comes from Schwyz.
But years later I read an article of a Swiss magazine which told that the Schweiz/Schwyz name link is a long running myth. Maybe Schweiz comes from Schwab an old Germanic region where our country stayed long time before 1291.

Is here someone that knows all this stuff?

The Swabian dukes supported the monastery of Einsiedeln with land donations. Later Schwyz expanded into Einsiedlean territory. Einsiedeln and the Habsburgians with their other affiliates became the main opponents of Schwyz until Schwyz won the battle of Morgarten in 1315. (Source)

From this point it doesn't make much sense that "Schwaben" becomes Schwyz.

However as your link states the term Schwyz for all three of the founding cantons was coined abroad and only adopted by the Swiss later.

And Uri and Unterwalden never got jealous because Schweiz became from Schwyz?

Oh I'm sure it itched them But like the other provinces of "Holland"/The Netherlands they got used to it.

In the end they consulted a distant ancestor of Tyler Brûlé who told them that the word "Schwyz" is an excellent start for a promising corporate identity.

Interesting... Noticed also the obvious similarity between the flags of Switzerland and that of the canton Schwyz?

Yeah, but the article I remember told that the first flag of Schwyz was all red without any detail inside.

When the Eight Canton Switzerland officialy adopted a flag (the one similar to our current flag), Schwyz still remained crossless. You can verify it at crwflags.com:
The Schwyz flag was originally an unadorned red banner, and the assumption that the modern Swiss flag derives from it is incorrect since the Swiss cross was in use by the Confederation about a century before Schwyz added it to its flag.
Source

The article (what a pity I lost the magazine!) was the same that told that Schweiz owes its name to Schwab.

Found that Bern might not be actually named after anything to do with bears at all, but more likely a corruption of the latin Brenodurum, which was a small Roman settlement in Teifenau next to the Aare (and also has a lovely bath house still there, or at least the bottom bit of one).
Myths are great, eh?

Gee, being I from Bern, I found it as disturbing as amazing.

Thank you for posting that one.