"Ap" means "son". The Welsh used to use a patronymic system, but surnames eventually fixed after Welsh union with England. Iceland continues to use a patronymic system.
From http://www.bbc.co.uk/wales/northwest/sites/familyhistory/pages/surnames.shtml
The Welsh patronymic system describes family trees in terms of the male line only and records the family association in the 'ap' or 'ab' prefix (ap is a contraction of the Welsh word mab, which means son). So, Rhys ap Dafydd means, in English, Rhys son of David.
Modern Welsh surnames such as Powell, Price and Prichard are the result of this contraction and a progressive tendency to Anglicise Welsh names: under the patronymic system they would have been ap Hywel; ap Rhys and ap Richard. The names Bowen and Bevan were derived in the same way.
Naming has a large arbitrary and cultural component and so is subject to the same structures as fashion. It seems for many centuries the only choices for Western Europe were Christian saint's names and for men mostly just the apostles.
But fashions change. It seems like in 19th c. Americans, the fashionable source was Old Testament prophets with z's and k's (Ezekiel, Zebulon, Zachariah).
And lately (early 21st c) American girls are named after president's last names (the aforementioned Madison, plus Kennedy, Reagan, Taylor (this is the answer to one of the questions). This is simply a current fashion trend mostly likely to be replaced soon from some new set.
One subtle pattern in English speaking areas for at least the past two centuries is to name a boy using the maiden name of the mother, usually a second or third boy since the first got the full name of the father (with Jr. or III appended).
As trends go, people sometimes follow the external view of the phenomenon (last names for first), which accounts for Jefferson Davis or (much later) Harrison Ford, whose mother's maiden names did not supply their first names. That these first names are the last names of famous people (American presidents) probably helped in their choice.
Best Answer
There's an interesting discussion on this topic in Wikipedia talk.
Their conclusion:
Go with tradition if you can find it, or with whatever manual of style you follow otherwise.