The one-time frontman of The Jam and The Style Council is suing Associated Newspapers at London's High Court for misuse of private information on behalf of daughter Dylan and twin sons John-Paul and Bowie.
He is seeking an injunction and damages, starting at £15,000 for each child, although counsel David Sherborne told Mr Justice Dingemans that the case was about vindication rather than damages.
He said that 55-year-old Weller, accompanied by wife Hannah, was there as the father of Dylan, who was 16 when the pictures appeared on MailOnline in October 2012, and the boys, who were then 10 months old.
"It is not his privacy claim. The claim is brought by three children of this father who just happens to be well-known as a musician."
Mr Sherborne said the decision to publish seven unpixelated pictures to illustrate a story about a "quality time" family shopping outing in California, was an unjustified infringement of their right to privacy.
He said Ms Weller, mother of the twins, had not been in the public eye before her marriage and had protected her children to the point of stopping her mother posting pictures of her first grandchildren on social media.
Associated Newspapers says the photos did not depict anything private or fall within the scope of Article 8 of the European Convention on Human Rights, which relates to respect for private and family life.
Counsel Antony White QC said: "They are entirely innocuous images."
The hearing continues tomorrow when Paul and Hannah Weller will give evidence.