My favorite would be “Four of Two” by They Might Be Giants (also often used as incidental music on This American Life.)
**Guster **has a great song called “Jesus on the Radio” that opens with the line, "5AM, March 16th… "
Daydream Believer, by The Monkees
Oh, I could hide 'neath the wings
Of the bluebird as she sings.
The six o’clock alarm would never ring.
And also a different “Four in the Morning” by a different Young, Jesse Colin Young. (He remade this song with The Youngbloods as well.)
Hooray! No one beat me to “8:05” by Moby Grape.
Blue Morning, Blue Day - Foreigner
“Out in the street it’s 6 am, another sleepless night.
Three cups of coffee, but I can’t clear my head from what went down last night.”
“Talking in Your Sleep” – Crystal Gayle
Three o’clock in the morning
And it looks like it’s gonna be
Another sleepless night
And another “three o’clock in the morning”:
“Chains of Love” – Big Joe Turner (by way of Johnny Burnette and the Rock and Roll Trio)
It’s three o’clock in the morning
And the moon is shinin’ bright
Two O’Clock Jump, a Harry James adaptation of Basie’s One O’Clock Jump.
Ain’t Misbehavin’, Fats Waller and others
“I don’t stay out late,
Don’t care to go,
I’m home about eight,
Just me and my radio.”
Five-O’Clock World – The Vogues
Clock on the Wall – The Guess Who
This is a GREAT pre-U.S. fame track featuring Burton Cummings at his punkiest. The song mentions, in order:
12 noon
Twenty to one
Quarter to three
Five
Twenty to nine
Twenty to ten
The midnight hour
I’ll always remember that one because of the time the spouse and I were driving, listening to the radio, and I asked him what time it was. “It’s four o’clock in the morning, damn it!” says Elton from the radio. 
No, we didn’t plan it that way. That’s why it was amusing.
*Well I looked at my watch, and it was quarter to one…
Well I looked at my watch, and it was quarter to two…
Well I looked at my watch, and it was quarter to three…
Well I looked at my watch, and it was quarter to four…
Well I looked at my watch, and it was quarter to five…
Well I looked at my watch, and it was quarter to six…
Well I looked at my watch, and it was way past six…
Well I looked at my watch, and it was a little past seven…
Well I looked at my watch, and it was quarter past eight…
Well I looked at my watch, and it was quarter past nine…
Well I looked at my watch, and it was somethin’ to ten…
Well I looked at my watch, and it was eleven thirty…
Well I looked at my watch, and it was straight up twelve…*
A few from my classic country/roots rock repertoire:
I’ve Got Five Dollars and It’s Saturday Night –George Jones, Faron Young and many others
Music and dancin’, flirtin’ and fun
It’s gonna go on till way past one
Move It On Over –Hank Williams
Came in last night a half past ten
That baby of mine wouldn’t let me in
Teenage Boogie –Webb Pierce
Saturday night about 8 o’clock
This old place begins to reel and rock
It’s Late – Ricky Nelson
If we coulda left home about a quarter to nine
We woulda had fun and plenty of time
Wake Up Little Susie –The Everly Brothers
We both fell sound asleep
Wake up, little Susie and weep
The movie’s over, it’s four o’clock
And we’re in trouble deep
And I don’t think anyone has mentioned “Rock Around the Clock,” which mentions several times in the course of things.
I’m not sure if it mentions a specific ‘time’, but Time by Pink Floyd must be the ultimate song about time.
“Ticking away the moments That make up a dull day Fritter and waste the hours In an off-hand…”
Sweet Bird of Truth by The The
“Six o’clock in the morning
and I’m the last person on this plane still awake”
Anything by Midnight Oil
“6 in the morning, police at my door…” – Ice-T
And its prequel: “Midnight, chillin’ at AM/PM…”
“Late in the Evening”-Paul Simon
Daysleeper - R.E.M.:
Receiving department, 3 A.M.
Staff cuts have socked up the overage
“7 A.M.” - Kesha
“Midnight City” - M83
“It was a Good Day” - Ice Cube (“woke her up around 1/she didn’t hesitate to call Ice Cube the top gun” and “two in the morning, got the Fatburger”)
Bob Dylan: “Joey”
“What time is it,” said the judge
To Joey when they met
“Five to ten,” said Joey
The judge says, “That’s exactly what you get.”
The Platters -"Twilight Time"
Southside of Bombay - “What’s the Time Mr Wolf?”