I am looking for what I think is a limerick from around the 1940's. It
sounds like Ona miori myick miann phyllison phollison niquish john
queasy quavy english navy stinklum stanklum be bi bo buck you're out.

goober..

I have to say, reading your question, I was intrigued for a couple of
reasons.  One, the wording of the phrase you are looking for and two,
the challenge of searching for a rhyme or limerick where the words
MIGHT be spelled correctly or might not.  Well, it worked and I found
your rhyme.  Let me explain how I found it as it was fun.

I tried a search first on the words I thought might have the best
chance of being spelled correctly, "be bi bo buck".  Nothing found. 
Then I tried "phyllison" knowing it might have been spelled with an f
instead of a ph.  Nothing on either spelling. Nest, I tried
"phollison" and lo and behold, a hit at
http://www.harpiesbizarre.com/spellbound.htm, which turns out to be a
Bewitched fan page and there was your rhyme, partially.  From there, I
searched under "qeeuvy quavy" and found your answer:

One-ry, or-ry, ickery, Ann! 
Fillison, folliso, Nicholas, John. 
Queevy, quavy, English Navy, 
Stinckelum, stanklum, buck!

another version is 

One-ery, two-ery, hickory, han, 
Fillison, Follison, Nicholas, John. 
Queevy, quavy, Virgin Mary, 
Stingelum, stangelum, berry buck

This is what is known as a counting out rhyme.  A counting out rhyme
is one that is used to choose someone for something.  One that may be
familiar is

one-potatoe, two-potatoe
Three potatoe, four
Five potatoe, six potatoe
Seven potatoe, more

Notice the rhythm with which the rhyme is said is very close to the
rhythm of your rhyme.

In doing the research, I came across a notation of Mark Twain using
this rhyme in an article written for the Virgina City Territorial
Enterprise.  This was a newspaper in Nevada where Twain wrote from
1862-1868.  You can see a reprint of the article at
http://www.twainquotes.com/18660200t.html

Thanks for your very interesting question.  If you need any further
clarification, please let me know.

Regards,

-THV

Search Strategy:
"phollison"
"queevy quavy"

References:
Counting Out Rhymes, Games
http://www.inquiry.net/outdoor/summer/counting_out/

Spellbound
http://www.harpiesbizarre.com/spellbound.htm

Twain Quotes
http://www.twainquotes.com/18660200t.html

Yet another site located at http://www.bigego.com/lyrics/bigwhoop.html
shows the following:

One riori
ickery ann
philiston pholiston
nicholas john
eavy quavy
english navy
stinkum stankum
bonk
ini mini
miney mo
catch a tigger
by the toe
if it hollers let it go
my mother said to 
pick the very best one
and it is why oh you

Apparently there are many variations of your rhyme.  :)

SEARCH PHRASE:
stinkum stankum (I recalled that variation from my childhood)

Lordy, you guys amaze me!! I hope someone gets five stars here -- it's
well deserved, if not for answer content but for patience. Hooo-hah.