When people are leveling and find a 3-man quest they might attempt to solo it. After all, with good gear and a little know-how, it's generally possible to take out an elite, and who wants to beg for help in General chat? But sometimes there are problems.
The Runed Stone Giants in Howling Fjord are rough on melee because of a stacking debuff they land on targets. High General Abbendis hits like a mack truck even though she's part of a 3-man quest. Gigantaur has a frequently used knockback that will punt melee all over the place and into other mobs. Alystros can't be rooted so caster classes are screwed as he pecks them to death.
Then there's Sarathstra, Scourge of the North, a 5-man undead frost wyrm that hits for about 1-2k on plate. As a balance druid she would be completely solo-able through Entangling Roots, since she's rootable, but not every class has that option.
Now enter one overly cocky holy paladin that has solo-ed every elite mob he's been given a quest to kill. High General Abbendis had been a close one, and it'd taken me a good 4-5 minutes to kill her, but after her and Gigantaur (damn that punting) the only thing that a 5-man frost wyrm seemed like it would take me is time.
And time it did.
Judging from the time left on my buffs, it took me about 15 minutes to get her to 20% where I could start using my Hammer of Wraths. After that the fight became imbalanced because some no doubt well meaning ret paladin decided to come by and "help" me, thereby ruining my challenge, but it was totally doable and I was never in danger of running out of mana. Another 2-3 minutes and Sarathstra would have been dead.
This is how to solo a hard-hitting elite in what will be a long-lasting fight as a holy paladin:
1) Judge Wisdom. As long as you're auto-attacking between spell casts (and you should be), mana will come back to you. I use Seal of Righteousness for added dps, but Wisdom could be good if mana becomes an issue. It never got to that point for me, not even after 15 minutes of constant combat. [Edit: Rohan suggests using Seal of Vengeance instead of Righteousness and I completely forgot about it! For a long fight where you'll get to build up all five stacks that's even better!]
2) Make sure Blessing of Wisdom is up. This does come in handy, and if the fight lasts long enough and you're not glyphed for it you will have to refresh.
3) Divine Plea every cooldown. This is key. Even if I knew I'd have to start healing myself in 2-3 seconds of hitting the button, I'd still do it.
4) If you are a blood elf, use Arcane Torrent every cooldown. It's more mana. I don't think it's necessary, but it helps.
5) Divine Illumination is nice if you remember to use it. I tend to forget, but even on a three minute cooldown I could have used it five times while fighting Sarathstra.
6) Don't hesitate to heal yourself. You can outlast just about anything with ret aura up as long as you have mana, but an unlucky crit could kill you if you're not careful.
On tougher fights I spent as much as 75% of my time healing myself and the rest keeping up judgements, refreshing seals, and throwing out offensive spells (Holy Shock or Exorcism depending). I try to Hammer of Wrath every cooldown once it becomes available, though healing takes priority. In a long lasting fight it will probably not be practical to use Shield of Righteousness just because the mana spent for the damage done won't be that high.
While I have bubble and Lay on Hands for when all else goes to pot, I found I didn't use either of them on Sarathstra. I did use Divine Shield on Gigantaur after an unlucky punt landed me with an add (I subsequestly kited him down a canyon where there was no possibility of adds), and Lay on Hands with Abbendis, but Sarathstra was pretty steady damage with no surprises.
The issue is really just to outlast. If I thought I was getting low I would budget my mana output, perhaps holding back on a Holy Shock in my dps rotation (since I would tend to go heal heal heal, then dps dps dps) and save it for the next round.
Other paladin specs don't have the luxury of being able to outlast. If the limiting factor of soloing an elite is living through the amount of damage being taken, a ret or a prot paladin probably isn't going to be able to hold out very long due to the size of their mana pools, though in full tank gear with Sanctuary up I imagine a prot pally could still do pretty good. Neither of them though, has quite the mana battery of a holy paladin.
Yes, it may take several minutes for a holy pally to kill an elite mob, but if you think about how long it takes for a guildie to fly over and help you, or how long it takes to find the right number of people in General chat, just killing the bastard as a holy pally could be just as fast or even faster.
I like not having to ask for help, or having to wait for anyone while leveling. I get to move at my own pace and still kill everything I have a quest for. Leveling holy isn't for everyone, but it's very, very good for being self-sufficient. Some might even say OP.