Inheriting the magical blood down the line, via a wizard father or a witch mother is understandable in case of Half-Bloods.
I am wondering as to how Muggle-borns end up with magical ability?
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The best theory I can suggest is that magical ability is like a recessive gene. If each Muggle parent is a carrier, they can produce a magical child. (However, this theory only works if you exclude Squibs :P)
Also, quoting JK Rowling:
"Muggle-borns will have a witch or wizard somewhere on their family tree, in some cases many, many generations back. The gene resurfaces in some unexpected places."
A study in the British Medical Journal concluded that while there is a genetic basis for magical ability it involved the interaction of several genes, some recessive. This explains why muggles can produce magical offspring.
The study addressed not just general magical ability but specific skills such as the ability to speak to snakes.
The study is here: http://www.bmj.com/content/335/7633/1305.full.pdf
As ykombinator wrote, JK Rowling has spoken on this issue herself.
I believe it is likely that there's a series of genes tied to having magical ability (plus some magic, I'm sure, since there simply CAN'T be a purely biological explanation).
It's probable that there are a series of genes that are linked together, and that many Muggles have 'damaged' versions of these genes - this could account for the various mediums, psychics, etc that exist in the Muggle world (those who aren't faking it, at least). If this is the case, a single random mutation in the gene could fix it for two muggle parents, or two muggles could have a child who gets healthy versions of the gene from one or more parents (the genes do seem to be dominant, since Muggle + Wizard invariably ends up with a Squib or a witch/wizard).
Some of the genes likely control the ability to see magical creatures (these must be fairly robust, since Squibs can see Dementors), some to the ability to control magic, etc. Squibs most likely come in various degrees of magical ability (or non-ability) as there's at least the perception that they can learn magic - otherwise the market for Squib magic teaching supplies would dry up quickly.
It could be:
a random alignment of junk DNA that winds up coding for magic sensitivity, as was stated before
that magic chooses and unborn/young child somehow linking with them
proximity to magic in use in utero or in early childhood
random chance (such as the existence of squibs in major wizard families)
Take your pick, I pulled the last three out of a hat. They all are based on random chance, and a small one at that.
In his notes on “Babbitty Rabbitty and her Cackling Stump” in “The Tales of Beedle the Bard”, Dumbledore mentions that the Department of Mysteries has studied how wizards get magic, and has found that wizards born to Muggles have a wizard somewhere in their family tree, possibly generations apart.
“As intensive studies in the Department of Mysteries demonstrated as far back as 1672, wizards and witches are born, not created. While the “rogue” ability to perform magic sometimes appears in those of apparent non-magical descent (though several later studies have suggested that there will have been a witch or wizard somewhere on the family tree), Muggles cannot perform magic. The best — or worst — they could hope for are random and uncontrollable effects generated by a genuine magical wand, which, as an instrument through which magic is supposed to be channeled, sometimes holds residual power, which it may discharge at odd moments — see also the notes on wandlore for “The Tale of the Three Brothers.”
- The Tales of Beedle the Bard
Therefore, Muggle-borns still get magic by inheriting the trait from someone in their bloodline, though in their case it’s from a more distant relative than their parents.
Magical talent, if it is indeed genetic, would be linked to a wide number of genes. Many of the most famous witches and wizards in the Harry Potter series are folk heroes who are well known as half human. Halfbloods not of wizarding blood and civilians, but of normal humans and some variety of magical creature. Merlin himself was a demigod, son of the celtic goddess of magic and one of her last mortal worshipers, a woman with no talent for the art. Cercie, babayaga, Morgan LeFay and many others, hell they even included angreboda, the mother goddess of the frost giants durring the 4th book.
Add to that the interview with JKR herself who said that there's really no such thing as muggleborns, but rather the multigenerational decendants of squibs the answer seems clear to me. You want magic ability you have to have some sort of magical ancestry.
Most of these types of questions assume that magical people have some special bit of DNA that turns them magical. In other words:
However, that only works when you look at humanity in isolation. When we take a look at the whole HP universe, we have to take into account that non-magical species outnumber magical species. Why? Additionally, we have to take into account that magical humans are only a small percentage of the human species, which doesn't make sense if magic was in any way beneficial, because evolution would have selected against non-magical variations in that case.
The only logical conclusion therefore is that magic isn't a beneficial ability. Which makes sense when you start looking at possible diseases. When you're magical, your body has to be able to defend itself against both magical and non-magical diseases. However, when you're non-magical, your body only has to defend itself against non-magical diseases.
In a way, the ability to perform magic is like colour-blindness. It's a disadvantage for the individual, but it can be beneficial to the population as a whole as long as it occurs only in a small percentage of the population.
So it should be the other way around:
One of the lessons of the book, illustrated by Hermionie, is that people rely too heavily on blood and genes to define their talents. People labeled as Squibs are still capable of basic magic it just doesn't come as easily to them.
I would wager that anyone is capable of becoming a wizard if given the time, discipline, tools and knowledge. Just like people are not just born with a given talent for math, people pick up traits through their life and growth. One who grows up in a wizard family will likely pick up wizarding skills, just like someone growing up in a house of mathmaticians probably picks up math pretty easily.
Nature vs. Nurture.
I believe that squibs are the reason for muggle-borns. Centuries ago, I would imagine that squibs would have been cast out of wizarding society. Those squibs would have mingled with the muggle population. Say, generations down the line, two different squibs decendents marry and have a child. That child would have a greater chance to produce magic because his/her genetical makeup still has the magical gene, though stronger than either of his/her's parents. I do wonder however if the child of a regular muggle and the decendent of a squib would be magical.
Another explanation, based on a real-world phenomenon called "trinucleotide repeat expansion": some genes comprise a number of repeating, three-nucleotide sequences (such as CAGCAGCAGCAG...) When such repetitive DNA string is copied, mistakes often happen and the number N of the "CAG"s will change between the generations. Usually N increases but a reduction is also possible, though rare. If N is less than a given value, say 25, the carrier is not affected (normal phenotype). If N is greater than, say, 100, the carrier will be affected (sick). In between, you may be affected or at least your children have a higher risk to be. So, let us assume that N is linked not with a pathology, but with magic.
So, for example: N less than 25 - Vernon Dursley;
N between 26-50 - a Muggle, low chances of having a magical child;
N between 50-100 - a Muggle, children may be magical or not (parents of Lily and Petunia);
N between 100-150: you may be able to perform some forms of magic, children in all probability magical; Squibs having some magic would be there;
N between 150-200: a wizard/witch;
N between 200-250: a powerful wizard/witch; N between 250-300: Grindelwald, Dumbledore, Merlin etc.
A few assumptions more: N more than 300: lethal, DNA becoming unstable, the embryo does not develop; thus the power has limits. This would explain why Mr Weasley said "without marrying Muggles, we would have died out". Squibs would be victims or rare deletions (N drops). As deletions are rare, Squibs would be less common than Muggleborns which seems to be the case. This model also explains why Muggleborns often have Squib ancestors: after the deletion, N starts re-growing, starting from a non-magical but relatively high value.