Runes are letters in a real-world alphabet Germanic in origin that predates the adoption of the Latin alphabet. There are a number of versions of these alphabets tied to different Germanic cultures and times, and the Elder Futhark, the Younger Futhark, and Anglo-Saxon Runes are commonly referenced. Cultures typically handed down the meaning and use of runes through poetry known as Rune Poems. These meanings were used to apply Runes not only for use in literature but also in divination rituals and magic.
By in large, magic in GURPS is handled through an advantage called Magery, which can be purchased in levels for character points. It accounts for the character's ability to cast magical effects, which are considered extraordinary compared to the average human. Different levels of Magery can be purchased to represent power and accessibility to certain spells. Spells are learned as Skills, which describe the difficulty for casting, the spell's affects, duration, range, prerequisite skills that the character must also have, and cost to cast. In general, GURPS magic does not distinguish on the style of play or source of magic. The spells available to a magic user are the same for a cleric, someone who receives this ability through the favor of the divine, a mage, someone trained to cast magic through ritual, or a sorcerer, someone who simply "knows" how to perform magic.
To be a Rune Mage in GURPS may simply involve describing your character as drawing runes, or calling out their names, perhaps using a piece of wood, a coin, or some other object that represents the rune while casting the spells. You would still purchase Magery and the spell skills as defined, and fulfill the prerequisite ladder to have access to more advanced spells.
The problem with this approach arises with how Dwayne wanted his character to use runes, to have a bag of inscribed runes from which he randomly draws at the time of casting, trusting in the gods to provide the tools to defeat his enemies. For example, the runic mage draws the three runes while in combat with a group of giants: Uruz (ᚢ), Raido (ᚱ), and Isaz (ᛁ). These runes might represent strength or the ox, travel, and ice. Perhaps the spell effect would be to call forth a giant oxen made of ice, who can charge the giants at exceptional speeds. The challenges here are interpreting the runes, then deciding what scope of affect the mage has ability or access to cast. If the mage has no experiencing in summoning, then Uruz might simply represent an increase in personal strength and speed.
So, we've raised two concerns: meaning and affect. Now, what about cost to cast? Access to individual runes themselves? Time to prepare? Action to activate? How do you translate the above action into dice rolls? Arg! A simple concept can quickly become complicated. Hmm... "trusting in the gods". Problem solved, it's GM choice! I do think there's a mechanism that can be defined here, but it will take some thought...
...or reference! After preparing much of this post, I decided to splurge and pick up Phil Masters' "Thaumatology" for GURPS 4th Edition. In it, he describes a Symbolic Magic system that uses runes in pretty much the same manner I outlined above. Cost is determined by what the rune represents: noun, verb, and scope. Affects are defined by combining runes, as listed above. Definitely worth it. It also turned out that Dwayne wasn't really interested in a random-draw casting, rather more structured. I think we have a good fit!
Another part of rune magic is divination. This type of use generally departs from the magic systems established by gaming systems. A divination example could be taken directly from many of the published Rune books in the Alternative section of book stores. There's almost no reason to further explore the ritualistic aspects of this for gaming purposes. Essentially, a diviner forms a question in his or her mind to ask the gods or goddesses and casts either a set number or the entire lot of runes within a circle, upon an animal skin, or other sanctified ground. Different casting patterns are used for different purposes or types of questions, and the runes are interpreted by the diviner. From a game mechanics point of view, the only thing important here are the Magery ability and a Divination skill.
All in all, I think a Runic Mage is entirely workable.