There's a song fragment bouncing around my head that I can't identify.
It's from the 50's and the singers sounds like Frankie Lymon and the
Teenagers but I don't think it's them. The chorus starts with the
phrase "Any old time of the week" and then goes on to list the days of
the week" a Monday, a Tuesday, a Wednesday and a Thursday..." it's an
upbeat, group harmony, doo wop song.

I am certain that you're remembering a 1958 song called "Every Day of
the Week," sung by a group called The Students. The lead singer's
voice sounded very much like the voice of Frankie Lymon. When the
record was first released, I remember kids in my school arguing about
whether or not this was a Frankie Lymon record.

Lots of other people also thought this was Frankie Lymon's voice:

"The Students
 Every Day Of The Week/I'm So Young 
 Released 1958 on Note
I first heard this great 45 in 1985 and I thought it was a lost
Frankie Lymon and the Teenagers song. I immediately started looking
for any Frankie Lymon record I could find (and afford), no luck at
all, was I ever frustated but it wasn't all in vain as I found some
cool Frankie Lymon sides I had never heard, did you know Frankie was
recording as late as 1968? neither did I but I actually found his 1968
45 on Big Kat, wow he had a deep voice and everything. But back to the
matter at hand, I had almost given up finding 'Every Day Of The Week'
when an R&B freak I knew came over and informed me it wasn't Frankie
Lymon at all it was a group called The Students!"

Head Heritage: The Students
http://www.headheritage.co.uk/unsung/review/219

You can listen to a clip of "Every Day of the Week" here (it's song 7 on Disc 2):

Oldies: Ultimate Doo Wop Collection
http://www.oldies.com/product/view.cfm/id_00422.html

The double CD which is available through the Oldies link above is a
wonderful doo-wop collection. I highly recommend it to any doo-wop
lover.

If you have RealPlayer, you can hear some great songs, including
"Every Day of the Week," on this site:

Central Oklahoma Classic Chevy Club: Doo-Wop Jukebox
http://www.55-57chevys.com/coccc/radoowop.html

I found a blog which states that the lead singer of The Students was
Frankie Lymon's older brother. While it's true that Frankie's brother
Lewis was a doo-wop singer, he was not involved with The Students.
Lewis Lymon fronted a group called The Teenchords. The lead singer of
The Students was Leroy King. You can read about the group here:

MARV GOLDBERG'S R&B NOTEBOOKS: The Students 
http://home.att.net/~uncamarvy/Students/students.html

Google search strategy:

Google Web Search: "every day of the week" "the students"
://www.google.com/search?hl=en&q=%22every+day+of+the+week%22+%22the+students%22

I hope this helps! If I have misidentified your song, or if anything
is unclear or incomplete, please request clarification; I'll be glad
to offer further assistance before you rate my answer.

Best regards,
pinkfreud

Yes, that's it. Thank you very much.