It does have that meaning, yes. It also has the meaning of name and monument.
Hence Ha'Shem does not mean "the reputation".
Ha Shem is just "the name." But shem does deal with His reputation... in fact, with God is goes beyond character, reputation and authority and extends to power and presence as well. Where were the sacrifices? Where His name was placed. But it also means what you said, name and memorial... consider...
Strong’s defines shem as name, as does Brown Driver Briggs, but both also define shem as a
mark or memorial of individuality. Both dictionaries go on to tie it to character, authority, honor, reputation and fame. In other words, in Western English especially that is spoken in the United States, deals with name as only what we call him. But in the Semitic mindset, shem might be dealing with what we call somebody, but it is more in line with the type of person the name bearer is. Please allow me show you what I mean.
In the Explanatory Notes of the 1998 version of “The Scriptures” (by ISR) we read this under this topic Name:
“The Hebrew word is
shĕm, which means much more than “name” in English. It is not a mere label of identification; it is an expression of the essential nature of the bearer of that name. In Hebrew, in the Scriptures, it speaks foremost of His authority. The Name of יהוה reveals Him, His character, the remembrance of Him, and His authority.”
To repeat, name in English is a label of identification, but shem in Hebrew is an expression of the nature of the name bearer. Shem reveals the person, his character, even his authority. This concept is supported by the Ancient Hebrew Lexicon (AHL), which defines words according the meaning of each Hebrew letter, it says this of shem:
“The wind, or breath, of someone or something is its character. Hebrew names are words given to describe character. The breath of a man is character, what makes one what he is. The name of an individual is more than an identifier but descriptive of his character or breath.”
But don't miscount reputation. If in Hebrew one's character is their name, then their reputation is also built on those same things. God has the reputation of being loving, kind, patient (etc.) but since those are His character attributes, then His reputation is built on character attributes and character attributes are His name.