My question is, why? I did the math, and as a first or second item it's only adding around 100 damage at most. For 3k gold that's quite unimpressive.
It's 100 AP + 100 damage, not just 100 damage.
To determine what that AP means I'll give a few averages/benchmarks I've found from comparing champions for the sake of comparison:
AP scaling on most champions is ~225% AP for their full QWER rotation, or ~450 AP for ~1000 damage.
Base damage is ~1000 @18 for pretty much anyone with a solid burst - from mages like Annie and Lux to tanks like Galio and Leona, granting ~55 damage per level. Base HP is ~2000 @18 for most champions, and mana is ~1000 @18 for most.
Anyways - Luden's 100 AP + 100+10% bonus ability offers ~330 total damage to a QWER as a first major purchase.
Rabadon's offers ~356.4 to a full QWER.
Lich Bane offers ~272.5 going off a base AD value of someone ~level 9.
Only a difference of 26.4 damage from Echo to Deathcap, and it's worth noting the party hat is 500 gold more expensive.
It's also worth mentioning most skills have the same (or very similar) mana cost from rank 1 to rank 5, which makes using a rank 1 skill when you have a rank 5 skill really mana inefficient for the purposes of harassing/poking/farming.
Why spend 50 mana to deal ~60 damage when you could spend ~60 mana to deal 200 damage?
In those cases - Luden's 100 base is likely going to out-perform the ~62 additional AP deathcap offers, as you'll generally need to cast ~3 abilities with deathcap to break even in overall damage: making deathcap mostly better at full QWER all-ins, and worse with mana-management, poke, and wave-clear compared to Echo.
Outside of end-game builds: it's also reasonably hard to 100%-->0% an opponent as a typical burst-caster. Usually you need to wear your opponent down a little bit, drop their HP to at least the ~70%~60% range first: which Echo is quite adept at doing.
