What can I use to effectively keep lures alive in solo eidolons?

I use the Amesha Archwing to set up a bubble around them

Really the best avenue is to have high enough stats that you can run through the Phases before the Eidolons can attack. the other options are all going to be switching to a different Warframe or Focus School, which will make it much slower overall.
Short of that, you can try not standing directly under the Eidolon so that if it stomps the ground it will deal a lot less Damage to the Lures. also Killing Vomvalysts rather than letting them shoot at the Lures.

If you don't want to change Warframes/Focus and those remediations aren't enough for you, increase your Damage more. move through both Shield and Health Phases in a single shot/hit.
Or, if you don't want to do all of the jobs at once, then don't play Solo. you don't have to, there's plenty of Players. if you're sure that you know what you're doing and you want to look for others that definitely do, you can always use Recruiting Chat.

I hunt Eidolons with Trinity for this reason. Yeah, it's slower, but it's the safe option; a Trinity built around her 4 is almost guaranteed to keep the lures alive.

I eventually took to not grabbing lures until the last limb. Enough little buggers rush up to fill the lures as you do the last limb, and the lures only take damage that short bit of time. Made it consistent to keep the lures alive. Probably not for everyone, but works for me.

cycle through limbs fast enough that you don't need to worry about them/just step aside if the eidolon slams and the lures will follow you/bring a healer like trinity or wisp

ideally you want to instantly break shield and limb so garry doesnt have time to throw out the light beams but if he does u just move them out of the way (easier said than done with their shitty ai but yeah). thats pretty much the only point lures are in danger of dying. you can drop ancient healer specters to make them take close to no damage but the specter itself dies super fast so youll have to keep refreshing it which really isnt worth it.

you can try Oberon

Smite adds radiation damage to your weapons, the grass protects you and lures during energy spikes . Renewal in bursts adds armor as well

In case you find yourself struggling during hydrolyst , the ancient healer specter also helps, but you might to have drop the specter often because it dies quickly

If you want to try out, Dante's overguard works on lures, tested it myself.

What build have you been using to do eidolons?

squad health restores work iirc

you'll want to be sure to park the lures and have good damage on limbs. shield too but i assume you got the meta.

Play as Trinity. if you are not looking to min max the hunt by doing as many as possible during the night then look for a setup that uses trinity.

I have used this setup in the past for example.

Easiest way is to kill the eidolon faster.

Actually, ancient healer specters might work

I use Nezha's Warding Halo augment to give them some extra protection