To Rose, it's Jack's song
The deleted scene is when we first hear it. The song is just a popular tune:
88 EXT. BOAT DECK - NIGHT
The stars blaze overhead, so bright and clear you can see the Milky Way. Rose and Jack walk along the row of lifeboats. Still giddy from the party, they are singing a popular song "Come Josephine in My Flying Machine".
JACK/ROSE Come Josephine in my flying machine, And it's up she goes! Up she goes! In the air she goes. Where? There she goes!
They fumble the words and break down laughing. They have reached the First Class Entrance, but don't go straight in, not wanting the evening to end. Through the doors the sound of the ship's orchestra wafts gently. Rose grabs a davit and leans back, staring at the cosmos.
ROSE (CONT'D)(pointing suddenly) Look! A shooting star.
JACK That was a long one. My father used to say that whenever you saw one, it
was a soul going to heaven.
The song was popular, and touched on a fantastical idea - to fly. The book America's Songs II: Songs from the 1890s to the Post-War Years, by Michael Lasser, explains:
A ride in an automobile was a possibility, but going for a spin in an air-plane was nothing but fantasy. Back when cars and planes were new, songs about them offered opportunities for young lovers to be along - a combination of freedom, privacy, and independence for a combination of adventure, love, and sex.
Within a song lyric, the two seductive invitations were pretty much the same: "Come away with me, Lucille, / In my merry Oldsmobile" and "Come, Josephine in my flying machine/ Going up we go, up we go". Alred Bryan's lyric combined a predictable invitation with a fantastical dream: "Whoa! dear, don't hit the moon, / No, dear, not yet but soon".
Later at the bow of the ship, Rose 'flies'. Jack is taking 'Josephine' on his flying machine. The song is now 'their song'. For the characters and the audience, this moment is connected through that song. The next time either the characters or us hear it, we're taken back to this moment:
ROSE I'm flying!
She leans forward, arching her back. He puts his hands on her waist to steady her.
JACK (singing softly) Come Josephine in my flying machine...
Rose closes her eyes, feeling herself floating weightless far above the sea. She smiles dreamily, then leans back, gently pressing her back against his chest. He pushes forward slightly against her. Slowly he raises his hands, arms outstretched, and they meet hers...fingertips gently touching. Then their fingers intertwine. Moving slowly, their fingers caress through and around each other like the bodies of two lovers.
Finally, at the end of the film, Rose sings it to her and Jack, who she thinks might still be alive. The reason is probably because to both of them, the song will take them back to time to the time when they 'flew', when they were alive, and when the possibility of flying to the stars was still real.
CLOSE ON Rose's face. Pale, like the faces of the dead. She seems to be floating in a void. Rose is in a semi-hallucinatory state. She knows she is dying. Her lips barely move as she sings a scrap of Jack's song:
ROSE "Come Josephine in my flying machine..."
ROSE'S POV: The stars. Like you've never seen them. The Milky Way a glorious band from horizon to horizon.
A SHOOTING STAR flares... a line of light across the heavens.
As Jack mentioned, a shooting star is when a soul goes to heaven, and soon after Rose realises Jack is dead.
The song was present at the start, middle, and end of their relationship, and as @Tetsujin pointed out, it connects characters and moments together for the audience.