Mr. Sandman, bring me a dream
As an appeal to Mr. Sandman, the singer is requesting to be granted a dream

Make him the cutest that I've ever seen
The singer desires for the dream man to have an adorable appearance

Give him two lips like roses and clover
The singer asks for the dream man to have red lips like a rose and four-leaf clover

Then tell him that his lonesome nights are over
The artist requests that Mr. Sandman delivers the message to the dream man that he won't be lonely anymore

Sandman, I'm so alone
The artist conveys to Mr. Sandman that she is very lonely

Don't have nobody to call my own
The singer emphasizes her loneliness and lack of companionship

Please turn on your magic beam
The singer believes that Mr. Sandman has the power to make her dream come true

Mr. Sandman, bring us a dream
The artist expands the request to include others

Give him a pair of eyes with a "come-hither" gleam
The artist desires for the dream man to have inviting eyes

Give him a lonely heart like Pagliacci
The artist desires for the dream man to have a sad heart just like the character Pagliacci in the Italian opera with the same name

And lots of wavy hair like Liberace
The singer wants the dream man to have an abundance of wavy hair like the famous pianist Liberace

Mr. Sandman, someone to hold (someone to hold)
The artist wants someone to hold, and repeats the request for emphasis

Would be so peachy before we're too old
Having someone to hold would be wonderful before it's too late and they become too old

So please turn on your magic beam
The singer requests Mr. Sandman's magical abilities once again

Mr. Sandman, bring us, please, please, please
The singer pleads with Mr. Sandman to grant the request for a dream for her and others