What is the name of Charlie Brown's dad's barber shop?

Are you sure it was ever named? I'm not. Charles Schulz grew up above his father's family barber shop, but I've been a "Peanuts" fan since my childhood in the late 60's, and I can't recall that a specific name was ever given to the shop.

Adults and their occupations didn't figure heavily in the strip, though the repeat strips in the Chicago Tribune are now talking rather extensively about Linus's beloved teacher, "Miss Othmar."

The Family Barber is as good as it's going to get, I think. Not mentioned in the comics it doesn't seem, but the real life inspiration perhaps.

Decades before O’Gara’s expanded into the space now known as the Shamrock Room in the early 1980s, the building was the center of the Schulz family universe: his father’s barber shop, The Family Barber

https://www.minnpost.com/stroll/2016/08/charles-schulzs-st-paul-little-pieces-schulziana-are-still-scattered-across-landscape

A Charlie Brown statue is called 'Family Barber Shop'. It is on display at the Associated Bank at the corner of Snelling and Selby in Saint Paul. This statue was sponsored by O'Gara's Bar and Grill, which is located across the street and on the site of the barber shop that was run by Charles Schulz's father. Charlie Brown's father also ran a barber shop, making this a natural theme. The artist is Duane Barnhart.

http://www.johnweeks.com/tour/peanuts/charliebrown.html