Jeanne Robertson.
For example:
JEANNE ROBERTSON VIDEO! Don't Bungee Jump Naked! - YouTube
How old are your nieces?
Bill Engvall is very clean as far as language and has very little risque material. He has a couple of (pretty tame) bits about sex with his wife, which is family friendly, right? 
If my neices/nephews were 12 or older I’d probably feel comfortable having them watch Bill Engvall.
Wow, took the two guys I came in here to mention. Oh, how 'bout Demitri Martin.
And Jeff Foxworthy. But if you watch the two of them on a Blue Collar Tour, skip Ron White. Hilarious, but filthy.
Agreed. Love Ron White but not family friendly!
I remember those albums well. He talks a lot about getting beaten by his dad as a kid, and uses a lot of corporal punishment imagery in discussing raising his own kids. All in a humorous, rather than tragic or brutal or ‘wrong’ sense.
“My job is the goalie. I kick them back into play.”
(During the bit about his wife laying about with a yardstick)
Jeff Dunham is funny, and I don’t think he’s over the top R rated…maybe some PG to PG 13.
His Drunk in Public bit had me collapsed on the floor, unable to breathe.
Aimed at adults? Sure. But if you think Ron White is filthy please don’t try Jim Norton.
Look up his uncensored stuff.
I’ve only seen what they’ve put on Comedy Central, and it looked mostly teenage kid friendly to me.
Well, upon seeing the thread title I thought “Gaffigan and Regan” but I see they were mentioned in the OP and about six times after that.
How about John Heffron? I don’t know if he’s 100% clean but all his work I’ve seen is.
Some of his uncensored stuff, besides the language, strays a bit into FreeRepublic Trump territory.
I guess times changed. I’ve taken a wooden spoon or two on the rare occasion that I’d take the blame for my brother’s mischief, but I never considered myself beaten or abused. shrug YMMV.
Bill Bailey doesn’t swear or work blue; a few drug references which will go over their heads, but otherwise his stand-up are fine for children and easily offended grandparents.
There’s a fair amount of stuff around from before he became a mega-star.
“This dick ain’t gonna suck itself.”
Funny as all get out, but still not family-friendly, which is what the OP was on about. I doubt there is a comedian out there who could make me wince. Norton is not even close. 
I feel like Eddie Izzard’s standups are pretty clean (unless you consider references to transvetitism to be inherently “adult”) and might teach you a bit of history.
Jim Carrey’s stand-up, from back in the day, was mostly just wacky, over-the-top goofiness. Reviews indicate that there are some sexual components, but I suspect that you could edit down and still have the majority of his routine left.
Granted, these are both by memory, so I might be wrong.
The times they are a changin’. To me I don’t know if I feel comfortable listening to a comedian who discusses child abuse like it is a non-issue with kids I feel responsible for.
My grandaprents watched that special with us and I never thought about it, but still. It bugs me thinking about watching that with kids.
