Regaining paladin features
See atonement, which has as two of its options
A paladin who has lost her class features due to committing an evil act may have her paladinhood restored to her by this spell.
Redemption or Temptation
You may cast this spell upon a creature of an opposing alignment in order to offer it a chance to change its alignment to match yours. The prospective subject must be present for the entire casting process. Upon completion of the spell, the subject freely chooses whether it retains its original alignment or acquiesces to your offer and changes to your alignment. No duress, compulsion, or magical influence can force the subject to take advantage of the opportunity offered if it is unwilling to abandon its old alignment. This use of the spell does not work on outsiders or any creature incapable of changing its alignment naturally.
Though the spell description refers to evil acts, atonement can also be used on any creature that has performed acts against its alignment, whether those acts are evil, good, chaotic, or lawful.
Note: Normally, changing alignment is up to the player. This use of atonement simply offers a believable way for a character to change his or her alignment drastically, suddenly, and definitively.
These are your rules-based ways of doing this. Atonement is a cleric or druid spell (also healer, shugenja, and apostle of peace, along with some domains and obsurer prestige classes), rather than a paladin one, so the paladin would need to consult a priest to get it done.
How to actually convince the blackguard to repent as necessary for atonement is a matter of roleplaying, and will depend on the paladin and blackguard characters themselves. What requirement the priest and/or god involved might assign (per the clause about using quest to assign penance) is up to you, as DM, and most likely would be tailored to the particular crimes that the blackguard has commited.
About those blackguard features...
Mechanically, RAW, a redeemed blackguard would actually keep his dark powers; there is nothing in the Dungeon Master’s Guide about ex-blackguards. Moreover, the rules in the Dungeon Master’s Guide for prestige class prerequisites state they only must be met in order to take one’s first level of the class, so not only could the redeemed blackguard keep his dark powers, he could actually continue to take blackguard levels.
This is mostly just a weird corner-case in the rules and probably an oversight, but I think there could be an opportunity for interesting roleplaying here. You see, actually using most of the blackguard features is going to cause problems for a paladin:
Many of the blackguard spells are [Evil] and thus casting them is an evil act.
Rebuke/Command Undead involves channeling negative energy, one of the few things the book explicitly lists as an evil act.
Associating with fiends is another thing a paladin definitely cannot do. That fiendish servant is thus off limits.
Smite Good, well, a paladin probably isn’t going to be attacking a good creature in the first place.
But at least Detect Good is OK! That one could actually be useful to a paladin.
Sneak Attack is somewhat dishonorable, but using it while flanking an enemy or when they lose Dexterity to AC because of something going on in the battle is probably just inside the bounds of what the Paladin’s Code accepts (attacking a completely-unsuspecting target probably isn’t unless there is a distinct war going on or similar).
Poison Use, on the other hand, is explicitly right out.
Finally, the Aura of Despair is not, apparently, optional: nothing says that a blackguard can suppress it. Even after redemption, he may just be stuck with it. He is thus branded for life and everyone who gets within 10 feet of him is going to know immediately that there is something very wrong with this fellow.
(Technically, Divine Grace and Dark Blessing stack, giving 2×Cha to all saves, since the bonus in each case is untyped; up to you if you want this redeemed blackguard to have ridiculously good saves or not.)
So here we have a redeemed blackguard, a paladin anew who has sworn to do no evil... with a fair amount of very-evil power at his fingertips. Can you say temptation? Especially if this man is going out into the darkness to do battle, there are going to be a lot of times when, say, it’d be really handy to have a fiendish minion, or turn the necromancer’s undead against him; a lot of cases where those things, properly controlled, could actually do a lot of good. But he has sworn off their use, and is going to be forever teetering on the brink of darkness. He’s not likely to get a second atonement.
And whoever his dark patron was, who gave him all this power: would that being rescind that power after the atonement, or just dig its claws in deeper? Maybe if the blackguard became a serious threat to that patron’s plans, his dark powers might get cut off (at the worst possible moment, of course), but otherwise I’d expect any fiendish power worth its salt to happily continue empowering a paladin that they’ve corrupted once before. “Whatever small acts of good are accomplished, will be forever tarnished when he falls again,” seems a pretty likely line of thought.
So I suggest that the blackguard, once he’s redeemed, continue to maintain his blackguard powers. Some of them are even usable, with care, as a paladin. But others will always be off-limits, available only at the price of his soul, and the Aura of Despair will follow him wherever he goes.