The word Tehillim is based on a 3 letter root →
הלל (HALAL)
Words with a wide variety of meanings are derived from this root:
Usually praising G-d (Elohim)(אֱלֹהִים), but could be also in reference to a man or a woman
To radiate, emit light like the sun or a lamp.
One interpretation of this meaning could be that the psalmist must clear up all darkness in his soul, so he can emit light while praising G-d.
On the other hand it could be interpreted as a desire to obtain light from G-d (to bring additional light into this world).
To shine in certain contexts can bring additional meanings - to make a show, to boast...
Boasting can be mild, somewhat positive, but when it becomes excessive, it leads to meanings like - to act foolishly, to be mad etc.
The depth of this root allows derivation of words with seemingly contradictory meanings (from praising to acting foolishly or being mad). One of the possible ways to reconcile it is the following idea: Tehillim bring Light to our souls, it leads to more praises, but when Light is absent, darkness takes over, which generates madness.
A large number of poems in the Sefer Tehillim contain the word Mizmor in the title.
The exact original meaning of this word is not certain.
The Hebrew root of this word is Z(ayin)-M(em)-R(esh)
1.ZEMER - song, melody, musical theme
2.ZMIRA - pruning, trimming, clipping
The above is not a complete list of possible variations based on this root, but the most relevant to this context.
The ancient text of this Book did not contain a title.
Rabbinic literature, based on etymology of the words, is using a title - Tehillim, "Songs of Praise",or Sefer Tehillim, "Book of Songs of Praise".
When this Book was translated to Greek (Septuagint),
in order to translate the word Mizmor, a Greek word - Psalmos was selected, which means - playing on a stringed instrument, or the song accompanied with a stringed instrument.
The English word Psalm is derived from the Greek Psalmos, therefore the Book is called - "The Book of Psalms".
The other meaning of the word Mizmor is based on the etymological meaning of trimming, clipping, which could be interpreted as cutting off a part of a whole. In a context of textual material, it could represent - a Paragraph....
But on a deeper, spiritual level, it could be interpreted as cutting off the darker side of our soul and in a resulting cleaner state to Sing Praises to God.