I found a couple of sites that say the older child's last name is Foley, but nothing really concrete about why they might have their mother's maiden name. I would guess that it is a matter of preference. When a woman divorces, she can choose to go back to her maiden name, and if she has custody, would probably want the children to have the same last name to avoid confusion. Perhaps in this case it is to keep them out of whatever celebrity limelight their father is in.
Dave married his now ex-wife, Tabitha Southey, on December 31st, 1991. They met at a bar when Dave was 20 and she was 17. According to Dave the relationship began right after being introduced when they argued over whether the lead vocalist of the group the Psychedelic Furs was a good singer. Tabitha and Dave were divorced in 1997 after 6 years of marriage. Together they had two sons: Edmund "Ned" Foley (born 1991) and Basil Patrick Southey (born 1995).
In 1983, he met Tabitha Southey in a vintage clothing store. Eight years later they were married. They had two sons, Edmund in 1991 and Basil in 1995. Unfortunately, the marriage didn't last and they were divorced in 1997. Foley married Crissy Guerrero in 2002. They have a daughter, Alina.