How did Saber become more linked to lions than dragons? We know Saber's real identity is Arthur Pendragon ("Chief-Dragon") and that she is linked to dragons through Merlin's sorcery:


  King Arthur was made into the incarnation of a dragon by Merlin by infusing her with the blood of a dragon, and one is shown in Realta Nua as a representation of her Magic Circuits and is visible standing near her Magic Core.


However, it seems that Saber is linked more to lions than dragons:


  Though she once felt more linked to dragons, she came to enjoy lions after taking care of a lion cub for a month at some point in her life. She claims that it is not that she "likes" them, but rather the happiness she experienced as he became attached to her brought forth a bond with them. He was very energetic, often biting or scratching, but she wished to be able to stay with him until the end. She has had feelings for them since then, and even remembers the fond memories while holding a stuffed lion. Shirou seeing the memory of her rubbing her cheek against the lion comments on it being a time where she looks like a girl her age.


If I do an image search on Sankaku Channel with the tags saber lion, I get 129 results, which is much higher than the 31 results that I get if I search with the tags saber dragon. A glance through the images reveals that, in addition to the stuffed lion in the original Fate/stay night, there are also Saber Lion, the Unit Lion race car that appears in Carnival Phantasm, and a grown lion often found to also accompany Gilgamesh and Iskander.

It does not appear that Saber taking a liking to lions is central to the plot, so why the stronger association of Saber with lions than good old dragons? Was it ever revealed (e.g. in an interview) why Kinoko Nasu chose to portray Saber more as a Lioness than a Dragon Lady?
The Lion reference can be dated back to her childhood. it's not mentioned on her page but Sir Kay's section. Sir Kay is the son of Sir Ector who Merlin left Arturia in the care of, that makes him Arturia's step brother though she had thought of him as her real brother (despite knowing the truth).


  Once while attending to her while she was sick, she told him that she would like to have a dream of a lion running on a grass field. He fashioned a poorly carved wooden lion that looked like a combination between a dog and a cat, and left it by her bedside. When she thanked him and told him that she had a pleasant dream of a young lion, Kay was shocked as he intended to make a decrepit old lion that would have been unable to hurt a rat. He had been worried that the lion might pounce on her in her dreams.


so while she is linked to Dragons, Lions was apart of her upbringing as the brother she loved carved her a lion when she was sick which most likely became a very cherished possession of hers and it's probably a subconscious thing of Arturia's extending from this. not entirely sure as to why she wanted to dream about a lion in the first place

I'm not sure as to how much of an impact this trait of Arturia's is on the story. Saber Lion is a parody much like how Magical Amber and other Magical Girls in the Nasuverse is a parody of Magical Girls. as far as i know it's not stated if the original Arthurian Legend depicts lions in the context of a connection with Arthur (the Pendragon Coat of arms normally depicts, as one would expect, dragons. According to Wikipedia Sir Tristran's earliest attributed coat of arms had lions however Kay's Lion would predate this) or if any derived works do or even which legend Kinoko Nasu may have used as a basis for Arturia

If i was to fashion a guess it is probably to make a more unique Arthurian Legend. Mordred is practically Incest from the most bizarre method, Nero, Frankenstien and Great Holy Grail War Jack are all women and most of the Hassan-i-Sabbah are described as not having faces at all so quite possibly while they all have a base real life legend Kinoko Nasu alters them slightly to have a bit for freedom in developing their backgrounds and legends to suit his needs.