The Chick: The White Magician Girl is classically female and always feminine. Tomboys, Sweet Polly Olivers, and Bifauxnen need not apply. Their femininity will usually be emphasized in her character design with the use of long hair and overtly feminine clothing, such as dresses or robes. Male analogs, which are rare, will mimic the same gentle appearance, sometimes to the extent of wearing long robes and long hair (or perhaps a suitably pious hat).
The Princess of Moonbrooke in Dragon Quest II is the Ur Example, debuting all the way back in '87. While she may have set a (possibly) unfortunate trend for RPG females, she's still a vast improvement over the Distressed Damsels that were so common at the time.
Ironically enough, she's more of a Red Mage than her successors since she learns the strongest attack spells in the game.
Rosa from Final Fantasy IV (also a Distressed Damsel) was almost a prototype for the breed, except her non-magical archery was often a match for some melee characters. She could wield a staff as well, but her best weapons were bows and arrows.
Garnet/Dagger from Final Fantasy IX uses magical "rackets" as well as staves. She fit this trope to a T until she starts getting her summons back, at which point she shifts to a more offensive-oriented character. Interestingly, this change is also marked by her cutting her long hair.
All these may be preceded by the White Mage of the original Final Fantasy, an early example of the character design, to the extent any gender can be determined. For a lot of young gamers in 1987, when Final Fantasy was released, the original White Mage party member appears to be gender neutral with a simple white and red theme and a hood drawn up, covering most of the face. Ambiguous gender, for a lot of players defaults to male. When your party "matures" and upgrades to prestige classes the White Mage's hood comes down, giving new perspective of the White Wizard with cascading red hair. This was taken by many to be The Reveal that the White Mage was a woman, much like recent contemporary Metroid's Samus Aran.
Breath of Fire often subverts the trope. Only the first and fourth games really play the trope straight with Nina, and even then Breath Of Fire IV gave her some offensive magic to let her qualify more as The Red Mage.
Fire Emblem: A fact present in several of these games is that one of your first allies will be a Cleric (on-foot female White Mage) or a Priest (on-foot male White Mage), and some rounds later you'll be able to recruit a Troubadour (mounted White Mage), and sometimes a Valkyrie (mounted Red Mage or mounted Magic Knight), a Sage (basically the same, but on foot and with either gender), or a Bishop (Light magic user and healer that promotes from Clerics and Priests (they can also promote from the all-male Monks, which use offensive magic instead of staves)).
Also in the first generation, Claude is a male example. Though he does have combat capabilities upon arrival as he is already a promoted High Priest, he doesn't have any offensive magic upon his arrival and has some of the most powerful healing staves, and can only use the most basic spells while being the only one who can use the most powerful staff.
In the second generation, Aideen's daughter Rana/Lana is the primary one, filling the same role as her mother, although depending on who her father is she has the potential to become more of a Red Mage after promotion.
Julia/Yuria, Diadora's daughter, is borderline. Much like Claude, she starts out with the capability to use offensive magic, but no offensive magic available for her to use. She's a bit better than her mother at holding her own in combat once she gets an offensive spellbook, but this can be attributed in large part to a spell that also heals herself upon a successful hit. And she's got the personality—like most, a carbon copy of her mother.
Nanna is another one whose qualifications are borderline, and perhaps in some ways dependent on her heritage. She's a Troubador, which means Sword and Staff, but due to her mother's bloodline she's at the very least guaranteed to get a decent Strength growth, which could be very good with the right father. Her mother doesn't give Pursuit, however, so with a father that fails to remedy this in favor of focusing on other aspects, she could qualify, especially with her Charisma ability which causes her mere presence to give a 10% boost to accuracy and avoid to all units within a three-tile radius. (In particular, if Lex is her father, his ridiculous Defense and HP growths combined with the Elite ability for quicker leveling up and lack of Pursuit turns her into a Mighty GlacierCombat Medic, able to deal decent damage if needed but more importantly able to take hits and stay near the front lines, healing and buffing the main fighters with Charisma.) Personality definitely fits.
Corple is a male example. Shows up late in the game at a low level and without any offensive abilities, but strong healing abilities (his mother also has the Blaggi bloodline, so he'll have a bonus to staff level). If Claude is his father, he'll be probably the best healer in the game.
Ellen (Cleric) is the straightest example. Clarine plays the role from a gameplay standpoint (Troubador) but lacks the personality (highlyTsundere, kind of a Rich Bitch), and Saul (Priest) could be a male example if not for his Handsome Lech tendencies.
Elphin the bard could be considered a male example. His role is strictly support (in gameplay, anyway; he also has plot relevance as he's actually Prince Mildain of Etruria), and he's got the kindly personality that his female counterpart Lalum (Genki Girl Dancer) lacks.
Cecilia may also qualify. She shows up later in the game and as such is already capable of combat as well (Valkyrie), but she's got the personality and is probably best used for support.
Priscilla, a Troubadour, is a sweet, kind-hearted girl. Her abilities are strictly support until she promotes to a Valkyrie, at which point she moves into Red Mage territory.
Ninian and Nils, the latter of which is a Rare Male Example, are interesting examples. They both definitely fit the personality type and neither has any combat skills at all. Their abilities squeeze in under White Mage for being purely support related, either giving a unit a second turn or providing a status buff, rather than any heals. They get extra points for being a pair of Mystical Waifs. Ninian is even The Hero'sLove Interest.
Natasha, a Cleric who joins up early in the game. Lacks any offensive magic at first, is a Defector from Decadence who hails from the country whose leadership is the antagonists, and quickly manages to win a powerful hired sword to switch sides.
L'Arachel (Troubadour), who is a little more Tsundere than typical but still fits nicely into the trope.
After Disgaea 2, Nippon Ichi games have any generic Healer unit. While the mentor system can be used to get healing magic from one person to another, the Healers are all female, and can wield staves (bows are useful too), and are always dressed in elegant dresses with long hair.
Also, the caring and compassionate part of this trope is played with (at least with the first Disgaea); if you read the description for that class, the generic healer's "elegant dresses" are actually lined with hooks as punishment, because the "caring and compassionate" healing arts are NOT the norm in daemon society...
MOMO from Xenosaga, who starts with a rod and in later games gets upgraded to an ether bow. Ultimately subverted in Episode III, where one of her two development paths turns her into an effective break attacker.
Elena of Grandia II is this. She wields a staff, does little in direct combat, has holy-themed magical powers, and is used mostly for healing spells. In fact, true to form, she shares a body with a Dark Magical Girl in the form of the Wings of Valmar.
Alice Elliot in the first Shadow Hearts game. She is gentle and demure, softening Yuri's harsher side. In combat, she uses a book and is the primary healer. She is also an exorcist to emphasize her holy powers.
Shana from The Legend of Dragoon, who wields a bow and is also a Mysterious Waif. Unlike the other playable characters, her dragoon form has a heavy emphasis on healing magic. Miranda later gains the same skillset and weapon, but her personality is too assertive to quite fit.
Fall-From-Grace in Planescape: Torment has the femininity, the nurturing personality and is the only healer found in the game; she only has about one offensive spell and is firmly a support character.
Shiho from Valkyrie Profile is the closest to fit the archetype. When she joins your party, she knows a large variety of healing and defensive magic, but no attack spells.
Rosea in Valkyrie Profile Covenant Of The Plume seems at first like a very generic example of the type, contrasted with the sadomasochistic fire mage Lieselotte. However, each of them blames the other for the murder that got them both exiled. In the neutral path, Lieselotte kills Rosea, speaking afterwards of how it was a fitting punishment for a murderer and a hypocrite. In the evil path, Rosea kills Lieselotte, sacrificing her innocence and becoming a fitting companion for your Villain Protagonist. In the good path, they both kill each other—and only in that path do you find out that neither of them committed the original murder!
Wynne from Dragon Age: Origins is an rare elderly example of this archetype, being around sixty when she first meets the Warden. Once she joins the group she quickly falls into the role of the resident Healer, frequently is shown to offer kind words of wisdom to various characters and becomes the unofficial Team Mom of the group. Aside from Morrigan, its clear that no-one has a bad thing to say about her. The other healing mages you encounter throughout the series are all men, but do not share Wynne's nurturing, maternal nature.
Utawarerumono has Eruru, who in the game serves as a healer during battles and in the anime adaptation still goes to battles and... apparently just watches without ever getting attacked really.
Kloe from Trails In The Sky — her orbment configuration is one line of mostly blue, meaning a high magic pool with healing and support skills. She also wields a rapier. Even her S-Crafts often involve healing.
Golden Sun has Mia, who plays this trope as straight as possible. Girl with long hair, wears feminine cloting? Check. Heals people, uses a staff? Double check. Has as her main (if not only) personality trait being caring and compassionate? Jackpot.
Goldmoon, from the Dragonlance series, is a prototypical example of this, the bearer of a powerful magical staff who sought to inspire the people to a return to faith in the gods. She is the High Priestess of her people. Goldmoon is a nurturing being and an encouraging and persuasive speaker. Her attire, however, is atypical, as she is a barbarian noble and cleric.