Origin
Stammler or Stemmler is the one who stammers. In the monastery of St. Gallen around 900 a. monk and poet Notker Balbulus, i.e. the Stammler. In Starkenburg/Hess. 1222 Gotefried Stamelere (Stemelere).
Nickname for the stutterer. The coarse folk joke did not shy away from describing the disability. Regional dictionaries are a treasure trove of similar swear words and nicknames, which have also sometimes become family names: Gänskragen (Geizkragen) = Stingy, Schilcher = Cross-eyed, Gratscher = broad and clumsy walker.
Family line
Unfortunately, I could only trace my family name in the family line back to my great-great-grandfather Gottlieb Stammler (1835-1889). He was born out of wedlock. His mother Anna Maria (1801-1877) was married to Friedrich Grether (1796-1834) and had three children with him. The marriage was divorced on September 9, 1833. Five months after the death of her ex-husband, Anna Maria gave birth to the "illegitimate" Gottlieb.