It would be odd to say
I am suing for children
I am suing XYZ for children
It could mean a number of things:
- I am suing because I want to obtain "children" as payment.
sue for something
to file a lawsuit in order to get something.
- If you so much as harm a hair on my head, I will sue for damages.
- Ted sued for back pay in his dispute with a former employer
- I am suing XYZ because I want children from XYZ
sue someone for something
to file a lawsuit against someone in order to get something.
- I will sue you for damages if you do anything else to my car!
- She sued her employer for failure to provide a safe workplace.
- I am suing on behalf of the children, who are unable to.
But this would be an ellipses of: "I'm suing XYZ for (all the) children (who are the victims)."
a. Used to indicate the recipient or beneficiary of an action: prepared lunch for us.
b. On behalf of: spoke for all the members.
c. In favor of: Were they for or against the proposal?
d. In place of: a substitute for eggs.
You definitely cannot sue on (ugly) children.
I am suing on children — NO
I am suing XYZ on children — NO
Mari-Lou A sues XYZ on ugly children — NO
(also impossible because any children of mine are beautiful)
But as @TRomano has pointed out in the comments, you can sue (someone) on the grounds of something. If an offence has been committed, you have the grounds = legal basis for a lawsuit.
Man sues wife on grounds of ugly children (headlinese)
Man sues wife on the grounds that his children are ugly
The preposition over in the headline cited by the OP is the most suitable one:
Man sues his wife over ugly children, and wins
The preposition is short for "on the question of", "in reference to", "concerning", "on the matter of", etc.
9. on the subject of; about: an argument over nothing.