Seymour's mother had a terminal illness and was dying. Also, she and her son had been banished to Baaj by Jyscal in the first place because of the uproar it caused in the Guado nation that he had taken a regular human for a wife. That was his way of keeping peace.
Because of who her husband was, and the Guado's own records of the past, she knew of the Final Summoning and that Yunalesca could turn her into a fayth. Since she felt that the people (Guado and regular humans both) would never accept her son, and since she was dying anyway, rather than leave him to face the world alone, she decided that the best thing she could do for him was give him the power to beat Sin, even if it would cost him his life. He'd be remembered and revered for doing that, but he didn't want her to do it.
When he saw Yunalesca turn her into a fayth and saw what she became, he fled from Zanarkand in horror (he was only 10 years old at the time), and he didn't return to Zanarkand to claim Anima as his aeon until 3 years before the game started. He then moved her fayth to Baaj and destroyed the entrance to the temple.
Techincally, Seymour had the power to beat Sin at any time, but he wanted to become Sin instead of letting his mother become Sin.
http://en.wikipedia.org/wiki/Talk:My...#Anima_article