The humorous American-English phrase to give the little lady a (great) big hand means to applaud a woman for an achievement. This phrase is based on the contrast between the adjectives little and big.

These are the earliest occurrences of to give the little lady a (great) big hand that I have found, in chronological order:

1-: From Service Talks, by T. O. Service, published in the Exhibitors Herald and Moving Picture World (Chicago: Quigley Publishing Company) of Saturday 17^th March 1928:

I herewith give the little lady a big hand. The little lady being Clara Bow ¹ and the reason for the applause being “Red Hair.” There’s a picture!

¹ Clara Gordon Bow (1905-1965) was a U.S. film actress.

2-: From the Los Angeles Times (Los Angeles, California) of Sunday 24^th June 1928:

THE BIG SHOW
The next great exposition, which Chicago will stage as its centennial celebration in 1933, will not be a show in industrial competition. Neither will it be a museum or a Coney Island. The bosses are telling that the show will be devoted to science and art. Many wonderful discoveries in various fields of science have never been brought to popular ken. These will be exploited as far as possible. Organizations in the arts, the sciences and the industries will cooperate and make general exhibits—thus replacing the customary displays by firms and individuals. The world will also be combed for its best paintings and for novelties in entertainment. It will be a universal panorama of the arts. If Miss Chicago can do that we will be glad to give the little lady a big hand.

3-: A punning use of the phrase occurs in the caption to the following photograph, published in the Los Angeles Times (Los Angeles, California) of Sunday 16^th September 1928:

“Give the little lady a big hand!” In robes of scarlet, trimmed with mink, and wearing her gold chains of office, Mrs. Lucia Foster-Welch, the mayor of Southampton, Eng., arrives in New York. Mayor Walker is shown greeting her at Manhattan’s City Hall. Pacific & Atlantic.

4-: From the review by Ann Silver of Queen of the Night Clubs (1929), published in The Brooklyn Daily Times (Brooklyn, New York) of Monday 18^th March 1929:

Since talkie debuts are just as popular now as cigarette testimonials, it was to be expected that Texas Guinan ², genial lady of the night clubs and padlocks, would screen and record her “Give the little lady a great big hand.” And you may be sure that La Guinan pulls all her favorite tricks and then some in this first talkie of hers, “Queen of the Night Clubs,” showing at both Strand Theatres this week.

² Mary Louise Cecilia ‘Texas’ Guinan (1884-1933) was a U.S. actress, producer and entrepreneur.

5-: A punning variant occurs in the caption to the following cartoon, published in The Montgomery Advertiser (Montgomery, Alabama) of Thursday 28^th March 1929 ³:

“GIVE THE LITTLE LADY A BIG HAND-FULL”

³ According to al.com:

March 1929: A period of heavy rain occurred Feb. 27-28 and March 4-5, then March 12-15, extreme rainfall fell across Alabama. Almost 30 inches of rain fell in Elba. Ten feet of water or more stood in the towns of Elba and Geneva. Thousands of people were stranded in Elba for up to three days before rescuers arrived.
Damage: $9 million, which would equal $120 million today.
Deaths: 13.

6-: From Billions of Tiny Insects To Fight Man’s Battles, by Frank Thone, published in several U.S. newspapers in April 1929—for example in the Tucson Daily Citizen (Tucson, Arizona) on Saturday the 20^th:

Meet Trichogramma minutum.
And give the little lady a big hand. For she’s a friend of yours, though you may not have known it. She’s hardly big enough for you to see her without a magnifying glass, but she packs a knockout wallop for many a one of the insect thugs that are constantly threatening man’s property and peace of mind.

7-: From the account of a tennis match, published in The Daily News (Passaic, New Jersey) of Wednesday 4^th September 1929:

The ladies’s [sic] singles was captured by Miss J. Callaway, who appears, on form displayed last Saturday, to be a coming Helen Wills ⁴. The gallery gave the little lady a big hand for her great work.

⁴ The U.S. tennis player Helen Wills (1905-1998) held the top position in women’s tennis for a total of nine years (1927-33, 1935 and 1938).

Among the numerous transferred uses of the phrase, an interesting one occurs in an advertisement for Perley’s Food Shop, published in the Cedar Rapids Gazette (Cedar Rapids, Iowa) of Friday 6^th October 1933:

Colonial Bread
Those Colonial loaves are beauties. Golden-brown crust—cake-like interior. If you haven’t tried Colonial, you’re missing a real treat. How such fine bread is produced at the price, is a mystery to us. Let’s give that little Colonial Lady a great, big hand! She symbolizes a lovely loaf at a bargain price.