There are two definitions for the Latin word 'anima' that could possibly apply to this character. Nature and soul. The one that I think best fits Lady Anima is, of course, soul.

According to the Webster's New Universal Unabridged Dictionary (one of those huge thing they have in the library), anima means -
noun
1. Soul; life.
2. a. inner personality that is turned toward the unconscious of the individual (contrasted with persona) b. the feminine principle as present in men (contrasted with animus).
Also means - breath, vital force, soul, spirit.

Animus means - 1. Strong dislike or enmity; hostile attitude; animosity. 2. Purpose, intention; animating spirit. 3. The masculine principle as represented in women.
Also means - mind, spirit, courage, passion, wrath.

Persona means - the mask, or façade, presented to satisfy demands of the situation or environment, and not presenting an inner personality of the individual; a public personality.

Note: all definitions concerning animus and persona are taken from the Psychology of C.G. Jung.

Let's look at anima first. Seymour's mother did give over her soul or life to Yevon to help him. Aeons are considered to be lifeless bodies with souls that live forever and eternally dream. She sacrificed it all for Seymour.

A personality that is turned toward the unconscious of an individual. This definition could be taken many ways, but there seems to be a certain way it is applied to Anima; Anima as an Aeon itself. Seymour's mother's pain was used as the will power for all of this, her pain and sorrow for Seymour because she knew he would never fit in. she was very sad for him and didn't want to see him not fit in like that, so she became an Aeons. Aeons are unconscious minds that eternally dream, and all of Anima's original pain and sorrow was force into the Aeon.

The feminine principle as presented in men can also mean many things. Anima was in fact female, and had sadness for her son. This sadness was presented in the aeon, but not completely. So Anima's sorrowful side can be considered the 'feminine side'.

Animus applies to the word Anima in a few ways. It, too, can be linked to the character.
Strong dislike, hostile attitude, animosity. When you hear those three, you can automatically think of Anima as an aeon. Seymour's mother was extremely disliked because she married and has a child interracially. In most societies, this is frowned upon, but when you're in love it doesn't matter. Both Seymour and his mother were rejects in all societies (Guado, Human, probably even Al Bhed), which developed into a sort of hate. After she became Anima, Seymour loved the power and became hostile with it. Seymour is like Anima's darker half.

Purpose, intention; animating spirit. Anima's purpose was to give Seymour something powerful and special that could make people respect him. Her intentions were good, but ended up being worse then she wanted when Seymour became like the other corrupt leaders of Yevon. Animating spirit is simply that Anima's soul can be summoned and her true soul can be shown in the room of the fayth in Baaj Temple.
The masculine principle as represented in women. I took this as being another part to Anima's purpose. Her masculine side could be a part of her that was enraged and angry that Seymour wasn't accepted into society, concentrated into her power.

Persona, like animus, also applies to the word 'anima' and the character on the story. The definition states that it 'satisfies the demands of the public' and 'doesn't present the inner personality of an individual'. This, once again, can be taken many ways, though I see it as Seymour and his mother's rank in society. They were both disliked because they weren't like everyone else, but when Anima was created it changed for both of them. Suddenly, no one cared that Seymour was half Guado, half Man; he was the leader of the Guado.

Other definitions that apply - breath, vital force, spirit, mind, courage, passion, wrath.