Does the name Yoshi from Super Mario mean anything?

According to ヨッシー in Japanese Wikipedia:

「ヨッシー」という命名は、任天堂社員のあだ名に由来。

(Translation: The name ヨッシー derived from a nickname of a Nintendo employee.)

So ヨッシー was the nickname of a real person in Nintendo.

Is this a common nickname in Japanese? Yes. Many Japanese surnames have よし (Usually 吉 in kanji, for example you may know this 吉田 and this 吉田), and ヨッシー is a reasonable nickname for someone whose surname contains よし. Also note that a nickname can be written in katakana regardless of the etymology.