I like this one! 
Slippin’ the Sausage since 1949.
What do you mean it’s not that kind of deli?!
Heh, thanks. 
Seriously, how about this as a t-shirt design:
Deli name across top
In middle, big line drawing of muscular arm hoisting a loaded sandwich (basically like a combo of this and this)
Underneath, in big letters: CHAZERCISE
I just threw up in my mouth a little bit.
Fill yer belly
at Cecils Deli
"Taste sensations for three generations"
Wait… why is this catchy? I mean, I get that it rhymes, but there doesn’t appear to be anything clever or even remotely memorable about it. What am I missing?
Tell your SO that I, at least, cast my vote for “Chazer-sized chutzpah”. 
“Kosher sausage, 12 inches long with a free bag of nuts.”
From Wikipedia:
During World War II, Katz’s encouraged parents to “send a salami to your boy in the army” which became one of the deli’s famous catch phrases,[3] along with “Katz’s, that’s all!” which is still painted on the side of the building. The former phrase is referenced in the Tom Lehrer song “So Long Mom (A Song for World War III)”, with the lyric “Remember Mommy, I’m off to get a commie, so send me a salami, and try to smile somehow”.
It’s a bit of a double fold for them.
Thanks, although I think it’s already been vetoed, just like my original idea of “Chome of the Chazer”. Not funny, said he.
Classics named Cecil:
The world’s smartest person, a seasick sea serpent, and St. Paul’s finest deli.
"We serve the worst sandwiches in town! Come & prove us wrong."
I really like this one. Sure, I understand the reference and many wouldn’t, but it’s not the original that matters.
It’s fairly short and not too trendy or slangy, which would necesitate changing it again later on.
Other than that, the one with the arm lifting the sandwich(Chazercise) would be my favorite.
I’m changing my vote!
“We were Here before you were Born.”
And we are still the wurst.
The key to the lox.
I vote for “Chazer me, bro!”
This American living in Indonesia does anyway.
If you choose my idea, can I have a shirt?
Some ideas…
Best of the wurst.
Deli neat, with a beef chazer.
Dill pickles, pastrami, salami, Reubens, chazers, blintzes, latkes, soups, cole slaw, and chopped liver. Are you sure you’re hungry enough?
Maybe it’s the rhyme salaaami with aaahmy? NY speech is often used to telegraph “funny, clever, cute.”
Since 1949!