They can take any race-restricted character option
As pointed out in this answer, in Pathfinder, most half-breeds of humans are also treated members of both parent races, and thus, also treated as humans. This rule extends to:
Orc Blood: Half-orcs count as both humans and orcs for any effect related to race.
Elf Blood: Half-elves count as both elves and humans for any effect related to race.
- Aasimars with Scion of Humanity racial trait:
Scion of Humanity: Some aasimars’ heavenly ancestry is extremely distant. An aasimar with this racial trait counts as an outsider (native) and a humanoid (human) for any effect related to race, including feat prerequisites and spells that affect humanoids.
Such a tiefling doesn’t need to succeed at a Disguise check to appear to be human and count as humanoid (human) as well as outsider (native) for all purposes (such as humanoid-affecting spells like charm person or enlarge person).
Such geniekin appear to be human, save perhaps minor features like unusual eye color, and they count as humanoid (human) as well as outsider (native) for all purposes (such as humanoid-affecting spells such as charm person or enlarge person).
Fetchlings with this racial trait count as outsiders with the native subtype and humanoids with the human subtype for any effect related to race, including feat prerequisites and spells that affect humanoids.
You count as both human and your race for the purpose of taking character options, such as archetypes, feats, spells, traits, and prestige classes.
Keep in mind that planar half-breeds exist for non-human races, as pointed out in Blood of Angels (Aasimars) and Blood of Fiends (Tiefling). Such variants would be an exception to this, as pointed out on the sidebar of Blood of Angels, for instance:
Non-human aasimars have the same statistics as human aasimars with the exception of size. Thus a halfling aasimar is Small but otherwise possesses the same statistics and abilities as a human aasimar-the difference is purely cosmetic. Non-human aasimars do not possess any of the racial abilities of their base race. However, they are usually raised in the same cultural context as other members of their base race, and thus generally adopt the same fighting style as their peers, use the same types of weapons and armor, and study the same skills.
So from a rules-as-written perspective, an Aasimar that has an elf parent and takes Scion of Humanity would count as humanoid(native) and still count also as humanoid(human) instead of humanoid(elf).
But are also affected by any race-specific effects
This may sound strong, but keep in mind that this also affects race-specific effects or effects based on creature type, like the Bane special weapon property, or an Intelligent Magic Item that wishes to defeat/slay all members of a specific race.
A half-orc, for example, is both a humanoid(orc) and a humanoid(human) creature. Meaning that they are affected by both a Human-Bane weapon and a Orc-Bane weapon (luckily, bane doesn't stack). A half-elf would meet the requirements of crafting a Cloak of Elvenkind. And so on.
The best resource for D&D elves, in my opinion, isn't any book at all, but rather afroakuma’s So You Want to Play an Elf. He is probably the foremost expert on D&D lore on the internet (at least, of those who don’t work for Wizards of the Coast), and his work incorporates decades’ worth of lore from the first, second, and third editions of D&D.
To be sure, his article on elves incorporates his own ideas not mentioned—at least explicitly, in these terms—in any D&D book. But they are extensions of the existing lore, and pains have been taken to make sure it is all official-compatible.
And, as mentioned, it doesn’t include material from fourth or fifth edition. Again, things are more-or-less compatible, and earlier editions had far more material than either of those do at this point, but some care might be necessary on some details if, say, playing with a 5e purist.
But the details are, honestly, the least important part, to me. And if your son has just started playing, there is a good chance he’s not a purist. So I think this article can be immensely inspiring for you.
In particular, the concept of short time and long time is the best idea I have ever seen for explaining elf mentality, and how their longevity affects them. It asks an elf to be more than just a snooty, tree-hugging human, but instead actually different. So that is the part I want to most highlight, though all of it is great.
The most crucial element to understanding an elf's mindset, though, is to understand long-time and short-time. The elf brain does not perceive time and receive information in the same fashion that most other humanoids do; elves live much of their adult life in a pseudo-trance known as long-time, in which they can idle away days or weeks at a time without devoting significant mental focus to anything. As creatures with a lifespan measured in centuries, elves have a lot of time available to waste on getting things right, and long-time is the biological mechanism that allows them to do it.
An elf is in long-time whenever there is nothing stressful that requires true focus. Elves in long-time can eat, drink, read, play music, craft woodwork or even hunt game or fight goblins if they are sufficiently at ease with the task at hand. A sort of mental auto-pilot, long-time supplements the elven need to trance by allowing elves to keep their minds at ease while developing muscle memory and picking up on subtle nuances and details that stand out as relevant. Elves in long-time can devote themselves to a given pursuit for anywhere from a week to several years without even having to notice what they are doing. Long-time is a fluid, waking dream; peaceful, functional and allowing the acquisition of experience over a long period of time. Elves can do many things while in long-time, but unless stressed, they will not be devoting the necessary focus to noticing much of what they're about.
This long-time is a deeper explanation for elves’ “harmony with nature” and how they can spend so long doing things that other races do more quickly, efficiently, and brutishly—elves naturally perfect their actions, not to maximize their effectiveness, but for grace and harmony.
It’s also why elves can seem so annoyed by other races—long-time tends to be impossible around those who cannot experience it.
The other crucial thing about elves that afro emphasizes—and this is more directly from D&D lore—is the fall of Araushnee/Lolth. That Lolth is a fallen elf goddess is well-known, but what is often forgotten is that she was the elf goddess of passion and destiny.
Now that she has reclaimed her divinity, Lolth still controls a twisted version of her old portfolio but rarely exercises it. The loss of Araushnee's portfolio has been a direct contributor to the stagnation and slow decline of the elven people, and one that Corellon cannot rectify. The only hope for the elves lies in finding those whose essence was not scoured - the few who were taken by Gruumsh when he stole away the Misty Isle.
This too goes a long way to explaining elves, their almost passive contentment, and their ceaseless hatred for Lolth and her drow.
Best Answer
Rules-as-written, they must pick one subrace of elf.
Rules-based answer: Pick one or the other.
According to Mordenkainen's Tome of Foes, the shadar-kai are a subrace of elf. This excludes half-elf and human as possible offspring. A half-elf is defined as the offspring of an elf and a human, while a human is a different race from elf entirely.
There are no specific rules for the offspring of two different subraces of elves. However, Player's Handbook, p.23, defines the following rule for playing an elf:
Mordenainen's Tome of Foes additionally adds three subraces:
Rules-as-written, then, the player must pick either the statistics of a high elf or a shadar-kai elf. Anything else, such as some homebrew mix of the two, must be ruled upon by the DM.
Lore answer: Canonically undefined
The lore of the shadar-kai as elves is thus far restricted to Mordenkainen's Tome of Foes. This is because shadar-kai were not canonically elves prior to 5th edition. They were described in 4th edition as humanlike, rather than elves.
In other words, the description of the shadar-kai in Mordenkainen's Tome of Foes is the sole source of lore on that race as regards their offspring with a high elf. That work does not define any such offspring, meaning that it is undefined.