Many restaurants sell baby back ribs, but why are they called baby back, and when was the first use of the term?
This is why they're called baby backs:
Baby back ribs (a.k.a. loin ribs, back ribs, or Canadian back ribs) are taken from the top of the rib cage between the spine and the spare ribs, below the loin muscle. The designation "baby" indicates the cuts are from market weight hogs, rather than sows.
N.B. emphasis added.
The Merriam-Webster dictionary gives the first known usage:
First Known Use of BABY BACK RIBS
With regard to the first recorded usage, a Google Ngram Viewer search for "baby back ribs" yields multiple matches from before 1980, including this mention in Sandy Lesberg, Specialty of the House: A Collection of Recipes from the Finest Restaurants Around the World Compiled with the Cooperation of the American Express Card (1970):
10 strips baby back ribs
2 tablespoons Hoisin sauce marinade
1 #10 can tomato puree
1 pint water to rinse can
4 ounces lemon juice
2 ounces soy sauce
I confirmed this publication date on Amazon, which still sells the book.
The oldest result that the Ngram Viewer returned is from Chicago Scene, Volume 4 (1963):
Dining is a delight at...DELMONICO'S Chicago's finest prime ribs, superb steaks, selected imported seafoods, barbecued baby back ribs...all carefully prepared and served in an elegant atmosphere.
Unfortunately, the snippets within the search result (as opposed to the one reproduced in the search result excerpt) aren't visible, but the date is consistent with the date that Google gives elsewhere for Chicago Scene, Volume 3 (1962).
Two other early mentions of "baby back ribs" involve ads from other Chicago periodicals: Omnibus and Chicago FM Guide, Volume 3 (1965) has restaurant listings for both The Colonial Kitchen in Highland Park (for "Barbecued Baby Back-Ribs, Full Slab, $2.55, Half Slab, $1.50") and The Flame (for "charcoal broiled baby Back Ribs at $3.50"); and Chicago, Volume 23 (1974) has a similar restaurant listing from an unnamed (in the snippet) eatery that charged $4.95 for "Baby Back Ribs Bar-B-Qued Beef or Chicken Combination." Jory Graham, Chicago: An Extraordinary Guide (1968) reports that
The menu [at Governor's Table] features Chicago Stock Yards steaks, a variety of fish, chops, chicken, and some of the finest barbecued Canadian baby back ribs to be found.
"Baby back ribs" also appears during the 1970s in city magazines for Madison, Wisconsin (from 1977); from New York City (from 1977 and later); and from Cincinnati (from January 1979 and later). Also noteworthy is a mid-1970s mention of "baby back ribs with Mexican sauce" in Anne Hardy, Where to Eat in Canada 1974/1975. Still, at least in Google Books' holdings, Chicago has the strongest claim to having popularized the term.