(Full) Drow make fairly poor spellcasters
The +2 Level Adjustment (LA) hits really hard, and the bonuses you get are just not worth it. You are a full spell level below where you would otherwise be, and that sucks. You also have lower saves, lower HP, and lower skills, all of which are really bad for an arena game.
Player’s Guide to Faerûn does provide a good solution in the form of “lesser drow,” which are drow with weaker stats but LA +0. The spellcasting you get back is easily worth the weaker stats. The “lesser” status, by the way, is purely a game-mechanical term, and does not (by default) imply any thing about your heritage or “drow-ness” in-character. It just means you are trading some of the drow stats for more class levels.
Multiclassing tends to be poor for spellcasters
Just as you don’t want to waste levels on LA because it weakens your spellcasting, you also don’t want to waste levels in other classes as a spellcaster. You want, as much as possible to have all the spellcasting you can get.
Lolth-touched might be worth it
Lolth-touched is an LA +1 template from Monster Manual IV, and it’s ridiculous: +6 Str and +6 Con, +4 Hide, +4 Move Silently, and immunity to fear. Usually the primary drawback is that you have to be, ya know, touched by Lolth, which is an unpleasant prospect in almost all situations. But if you’re playing a female cleric of Lolth... you may be able to do it.
Still might not really be worth it since it doesn’t really improve spellcasting, though clerics are good in melee as well.
Archivists, clerics, druids, and wizards are the most powerful spellcasters
The ability to prepare new spells each day is incredibly flexible and powerful, and these classes provide the strongest spell lists with that feature. Lesser drow do not get any mental ability score bonuses, which make them neutral with respect to which of these is best, and female drow are traditionally clerics of Lolth. Any of the others is an option, but I’m going to assume cleric because it’s most fitting.
The only other thing I’d consider here is going beguiler/wizard/ultimate magus. Beguiler is a decent spellcaster from Player’s Handbook II, and very fitting for drow: it’s all about stealth and subtle magic, lots of illusions and enchantments. Ultimate magus from Complete Mage allows you to advance a prepared arcane class (like wizard) alongside a spontaneous arcane class (like beguiler), and entry is relatively easy. Recommended build is Bglr 1/Wiz 4/UM 10, at least until 15, and take Practiced Spellcaster (beguiler) from Complete Arcane at 6th or earlier to ensure you get maximum wizard spellcasting.
Ultimately, the cleric is probably more appropriate as a female drow, and probably stronger anyway, without the lost spellcasting level, but I wanted to mention it as a really solid drow option. If you are required to use the full drow stats instead of the lesser stats, this build also makes much better use of the +2 intelligence. You just may have to explain why your female drow has taken such a stereotypically “masculine” path.
Speaking of being a full drow, if you are and have +2 intelligence, but want to remain a divine caster, consider archivist instead of cleric. The loss of Rebuke Undead is a shame, since it means you cannot use some of the juiciest divine feats, but the intelligence bonus may make it worthwhile.
In general, the really most powerful feat option for a cleric is Divine Metamagic and Persistent Spell. These allow you to exchange Rebuke Undead uses for reduced metamagic cost, and therefore get your powerful Personal buff spells to last 24 hours. Divine power and righteous might are two great choices, for example.
Nothing super-special here; use the typical cleric power-houses; read the cleric handbook if you’re unfamiliar. Again, divine power and righteous might tend to be the best available.
Playing to stereotypes: poison
Poison is expensive and difficult to use, but a cleric of Lolth, in particular, is well-suited to making the most of it. The feats Lolth’s Caress, Master of Poisons, and Poison Spell are all found in Drow of the Underdark, and allow you to apply poisons quite liberally. Use of minor creation et al. can provide cheap poisons, and if you’re allowed, Wild Cohort can provide a snake or spider pet that can provide poison, even if you never bring it into the arena.
Lolth’s Caress is particularly brutal: expend a use of Rebuke Undead to force anyone suffering from poison to save against its secondary effect right now. For more on using poison effectively, see the Arsenic and Old Lace Handbook. Poison tends to be way better at low levels.
Traditional solid cleric prestige classes are probably your best bet; none of the drow-specific ones are all that impressive. Bone knight (Five Nations) and ordained champion (Complete Champion) are particularly strong melee-oriented prestige classes, and would go well with, say, Lolth-touched. Divine oracle and contemplative (both Complete Divine) are awesome for simple, sweet class features.
If you are not a cleric, you might try to sneak your way into sacred exorcist (Complete Divine), as it grants Turn Undead. It’s a Good class, though, so that may be problematic for you. Only bother with this if you are going for specific divine feats, like Lolth’s Caress, Divine Defiance (Fiendish Codex II), or Divine Might (Complete Champion).
If you can waive the god-specific requirements, or for some reason make a drow cleric of Wee-Jas, the ruby knight vindicator from Tome of Battle, after a dip in crusader from the same, is awesome. Ideally, go Clr 4/Cru 1/RKV 10.
For more ideas, see the cleric handbook