In the Irish poem The Four Farrellys by Percy W. French is the line
Slainthe, Slainthe, Fransheen.
What is the English transation?

Hello.


"Slainthe, Slainthe, Fransheen" translates approximately to "Cheers,
cheers, little Francis."


Slainte is a Gaelic toast meaning "good health!" (i.e., it's the
equivalent of "cheers").
See: 
Useful Words & Phrases in Gaelic:
http://shenanchie.tripod.com/irelandtwo.htm#wordsphrases
Encarta: Slainte
http://encarta.msn.com/dictionary_/Slainte.html


In Irish Gaelic, the suffix "-een" means "little."

"Almost any word that ends with -een  can be assumed to be Irish and
smithereens is no exception.  The -een  suffix represents the Irish
diminutive -?n as in colleen "a young girl", boreen "small road" or
"lane", kippeen  "a small stick"."
source: Takeourword.com
http://www.takeourword.com/Issue050.html

As such, "-een" is added on to names to denote "little."

You may have heard of Irish names such as Maureen and Cathleen (or Caitlin). 

"Caitlin means little Kate
Maureen little Mary 
source: Geneology Names
http://experts.about.com/q/2028/1341794.htm

The "-een" suffix may also added on to boys' names to denote "little."
 For example, the name "Michaeleen" means "small Michael."

Here's an example from the movie "The Quiet Man" (1952):
" Sean: I'll tell you why Michaeleen Oge Flynn, young small Michael
Flynn who used to wipe my runny nose when I was a kid. "
http://www.filmsite.org/quie.html


Thus, Fransheen would be "little Francis." 

 This is appropriate in the context of the poem because the character
being addressed is named Francis:

"Oh! If you're that Francis Farrelly, your dreams have not come true,
Still, Slainthe, Slainthe, Fransheen! for I like a man like you!"
http://www.kraffe.org/kraffe/liam/diary17.htm

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search terms:
slainte cheers
een suffix little gaelic

I hope this helps.  Slainte!