I came across a book, Surfing: the Spirit of Hawaiian Kings, with lots of information about women’s surfing. (The female inclusiveness is pretty untypical for a 1966 publication.) The book names several oral traditions about women surfers, as well as accounts by European naval officers and missionaries from a time when Hawaiian culture remained intact. Some excerpts follow.
In the first European account of Hawaii, Captain Cook described how a princess” (meaning a woman of the ali’i – other accounts call such women “chiefess”) “paddled her board through heavy surf to catch an ride the rolling waves,” and observed that women were just as ready “to place themselves on the summit of the largest surge.” 
Many later accounts emphasize that women were active surfers in eastern Polynesia. Finney and Houston say that In the western islands, adults didn’t surf, and that boys surfed.) But they say that “…both sexes rode waves on tiny Rapa, south of Tahiti. In Hawaii everyone enjoyed it, men and women, young and old… The same was true of New Zealand and among the Marquesans, northeast of Tahiti.”  Morrison, one of the officers on the famous British ship Bountycommented of Tahiti, “at this diversion all sexes are excellent… the children also take their sport in the small surfs.” 
But the most extensive development of the sport and art was in the Hawaiian Islands, where all genders, ages, and classes took to the waves. On Hawai’i itself, the missionary Ellis remarked that when good surfing waves started running, “the thatch houses of a whole village stood empty; daily tasks such as farming, fishing, and tapa-making were left undone whie an entire community—men, women and children—enjoyed themselves in the rising surf and rushing white water.” 
Oral tradition recalls that king Kamehameha and queen Kaahumanu surfed side by side…”  Chiefly surfers had praise-singers who sang special songs for their surfing. Certain surfing areas were reserved for the aristocratic ali’i. On Maui, the surfboard was “an article of personal property among all the chiefs, male and female, and among many of the common people,” according to C. S. Stewart (another missionary, the place was plagued with them) in 1823. [38, 35]
Finney and Houston are refreshingly informative about female participation in surfing. “Although early accounts do not mention who surfed the most, eye-witness descriptions usually refer to adult men and women, with an occasional mention of children riding small boards close to shore. Except on those beaches where the most dangerous swells peaked, men and women shared surfing areas equally.” 
And again: “a large percentage of wahines of early Hawaii was skillful surfers, and sometimes champions.” Early engravings show women riding surfboards. As a 19th century publication called Thrum’s Annual related, ‘Native legends abound with the exploits of those who attained distinction among their fellows by their skill and daring in this sport, indulged in alike by both sexes; and frequently too… the gentler sex carried off the highest honors.” 
One of the best surfing spots in Kou was Ke Kui o Mamala in the harbor of Honolulu: “The break was named after Mamala, a famous woman surfer and prominent Oahu chief. According to legend she was first married to Ouha, the shark man…” But Mamala left him for Honokaupu, the owner of a nearby coconut grove. Ouha then cast off his human form to become the shark god of the coast between Waikiki and Koko head. A song was sung about this love triangle. 
Another surfing love story involved a supernatural woman who lived in a cave overlooking the North Shore of Oahu. A prince of Kauai’I came to surf in the renowned, colossal breakers at Sunset Beach. A Bird Maiden who lived in a cave above watched him surf, then sent her bird messengers to bring a lehua lei to him and bring him to her. They spent months together as lovers, until surfing season came around again. He promised to never kiss another woman, but broke this vow when a woman came up after his surfing and put an ilima lei around his neck. The birds brought word back to their mistress. She ran to the beach, tore off the ilima lei and reclaimed him with her lehua lei. The Bird Woman ran back to her cave, with him in hot pursuit. He never caught her, but turned to stone halfway up the mountainside. [39-40]
Surfing had definite romantic connections. If a man and woman happened to ride the same wave together, “custom allowed certain intimacies when they returned to the beach. More formal courtship was also carried out in the surf, where a man or woman tried to woo and win a mate by performing on the waves.” 
If a man and woman happened to ride the same wave together, “custom allowed certain intimacies when they returned to the beach. More formal courtship was also carried out in the surf, where a man or woman tried to woo and win a mate by performing on the waves.” One story tells how Kauai’i champion Hauailiki courted an ali’i woman at Keeau, Hawai’i. But she was unimpressed by his surfing, so he resorted to a display of body-surfing. She called him to her and placed a lehua lei around her neck, as it was her custom to recognize good surfers—but he got no farther than that. 
Another story reflects more coercive behavior. Kelea, sister of the king of Maui, “was famed as the most graceful and daring surfer in the kingdom. One day while surfing at Lahaina, she accepted an invitation to ride in the canoe of a visiting chief from Oahu. A sudden squall swept them out to sea, and taking advantage of the storm, Kalamakua abducted Kelea and sailed for Oahu. He presented her to his high chief Lo-Lake. Kelea was furious, but she was a captive. To make things worse, this “husband” liked to live inland, well away from Kelea’s beloved waves. She vowed to return to Maui. The story says that she stopped to surf at Ewa, and wound up accepting Kalamakua’s proposal of marriage. 
Kapu (taboo) was used to exclude commoners from some surf areas. Some accounts describe ali’i and commoners surfing in the same areas, but this may date from after the kapu system was broken in 1819. At least one fine surfing place was reserved to a single ali’i woman: “a special surf at Waikiki that was taboo to everyone but the Queen. For riding to shore on one of the royal lady’s waves, one young man was severely beaten and nearly put to death.” This story recalls what was said earlier about customary connections between sharing waves and sexual intimacy, and now it seems to intersect with kapu against men of lesser rank being sexual with ali’i women. 
Queen Emma had a special surf chant. The chiefly class of ali’i had praise-singers who chanted songs for their surfing. [46, 38] Another story turned on an old woman who was the personal chanter for Naihe, a champion of Ka’u, Hawai’i. His enemies were conspiring to kill him, and secretly agreed that no surfer would return to shore until their chant had been sung. He didn’t know, and neither did his chanter. She fell asleep, not expecting to be called on, and he was forced to remain at sea. But one man woke up the old chanter and revealed the plot. She ran to the shore with streaming tears to sing the song, and saved Naihe.  Oral tradition may name her, but these authors do not.
Huge changes took place when Europeans colonized the Hawaiian Islands. Even before the coup that seized power from the Queen Lili’uokalani in 1893, the missionaries had already wrought havoc on Hawaiian culture, in every sphere of life: dress, ceremony, dance, and culture. Surfing was no exception, and came in for repression on grounds of “scanty costume,” “immorality,” sexual freedom, and its connection to to the Aboriginal religion. [62-4] It languished in a climate of fear and cultural lockdown.
Tahiti underwent similar cultural colonization, and by the late 1800s had abandoned surfing and swimming, along with dance, games, and their old religious customs. Instead, according to this book, their only entertainment was singing Christian hymns. [62-4]
A revival of Hawaiian surfing was underway in Oahu by 1907. Native Hawaiians had to band together in a club to get access to the fabled beach in Waikiki. But the interruption of the surfing tradition, along with all the other cultural changes, had cut down women’s participation severely. It took many decades for women to regain some of the ground they had lost. Today, women surfers are riding the waves again, as well as women open-sea rowers like those who race from Moloka’i to Oahu every year.
Surfing: the Spirit of Hawaiian Kings. Ben R. Finney, James D. Houston. Rutland VT: Charles E Tuttle, 1966
(The book does not identify the engravings shown, but they would be from the early to middle 1800s.)