I would use it for the BC levels as well. The quest hubs in BC typically have 50% easy quests that can be completed very quickly (like kill a specific monster or get ONE item) while the other 50% can take much longer (i.e. collect 30 of a 30% drop rate item from slow spawning mobs.) If you can reach the appropriate level for the next quest hub using the elixer to skim the easy quests, then BC will be as painless as possible.
Or as flyingbacon suggested, you can just chain run BC instances and not be bored.
I'd use it to skip whatever you hate most. Many people dislike outland, other people dislike northrend.. take your pick. Fill your quest log with a couple of quest hubs, pop potion turn everything in is a good start to the hour.
Chaining dungeons with a good tank and good dps is obviously the most efficient though.
Whatever content you hate most? Personally I used it during WotLK for two reasons:
1) BC can be mostly leveled through dungeons and since I was a monk I just rolled around as BrM and chain dungeoned and did a few quests.
2) Wrath quests are very well grouped up in nice camps. Using the Elixer out here I was able to pump out a lot more quests in that time span than I could in BC and makes the grind up to 80 a little less painful (as opposed to only needing to hit 68 in outlands).
Personally I absolutly HATE northrend so I lvled my DK up to 72, finished around 12-17 quests (can't remember now, was like half a year ago) and DIDN'T hand them in. I also had my husband helping me and waiting for me with 2-person mount.
I also changed all my spells to the one that will help me survive and changed spec to blood so no random aggro could 1-shot me. Then I flew the village (in dragonblight) where I had most of my quests (had my husband waiting for me with the mount), popped the elixir, turned in all the quests in that village, quickly hopped on the mount, flew to the next village where I turned rest of my quests and hopped on the mount again.
Then my husband boosted me through 5 instances (can't get more in 1h anyway) that were highest possible lvl for me but still had quests available. That way I went from 72 to 80 in 1 hour
Just watch out for some random aggro so you wont die and lose the buff!
Elixir of Ancient Knowledge
Item Level 1
Binds to Battle.net account
Use: Increases experience gain by 300% for 1 hour. Does not work for players above level 84.
"A faint whispering can be heard from the bottle."