So, if there is a full melee army (0 archer), which wraith to choose? I got inspired to this question by making right now an army list for dragon.
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Shadowlord: Enemy archers are pain for melee forces, so making them neglible might be a good idea - force your enemy to melee, and you will win (unless he is dwarves, then it is gg wp). This would probably be good against elves, rangers, etc?

The problem with shadowlord is next: As it makes enemy archers neglible, it makes a question: WHY ARE YOU NOT ARCHERING YOURSELF, instead of wasting this bonus by walking into enemy lines? I don't know if it can be after all a good thing to have a shadowlord in melee army?
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Dwimmerlake: If you do not have any magic to protect your unit, let's say you have a 250 points dragon, then an enemy mage could be a problem. And in this case, as odd as it sounds, the Dwimmerlake might be the bet. Mathematically enemy will be able to use only 50% of his points. So, enemy wizard will fail a lot of spells. Also, enemy with a lot of might looses his advantage!

Cons: Dwimmerlake has no might, but neither will have enemy after this ghost enters battlefield. It has a lot will and faith so it can play bravely. I cannot see anything specific to blame? Ringwraiths are taken to counter strong enemy heroes and mages, and this is exactly what Dwimmerlake does?... Better than any other Wraith?
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Well, it depends on what else you've got in the army. I'm going to assume you're thinking about the most recent one you posted in the army help section:

*Dragon
-8 morannon orc, 1 item each
-1 warg rider, shield and throwing spear
-2 giant spider

*Dwimmerlake
-8 morannon orc, 1 item each
-2 warg rider, shield each
-2 giant spider

*Orc Shaman
-8 morannon orc, 1 item each
-2 warg rider, shield and throwing spear each
-2 giant spider

When you're thinking which Wraith to use, I like to break it down as follows:

-Witch King:
Depending on how expensive I want him to be, do I need a medium cost hero-hunter (Morgul Blade), do I need a combat machine (full M/W/F with the Crown, mounted on something)?

-Khamul:
Do I have the points to put him on a Fell Beast? Do I need a strong combat hero?

-Dark Marshall:
Would my warriors benefit from his banner (ie do I have a high Fight Value/Shade)?

-Knight of Umbar:
Same Fell Beast question as Khamul. Am I facing BIG heroes?

-Undying:
Do I need a leader? Do I have more magic around me?

-Shadow Lord:
Are my warriors squishy?

-Dwimmerlaik:
Does my opponent have a lot of heroes? Do I have low Might in the army? Do I have models that need spell protection?

-Tainted:
Am I facing a low Courage army? Do I have a Shaman? Do I need Heroic Moves with this army?

-Betrayer:
Do I need an all-in-one hero killer and can I put him on a Fell Beast? (Compared to what I described in your Shelob topic, where you need a lot of pieces to make her work)

-Budget Wraith:
Do I have 60ish points to spend?


Of these, I would agree with you that your choices are probably down to the Shadow Lord or the Dwimmerlaik. The army list you put up is in an awkward spot of needing both: you have a lot of vulnerable or low Defence models that would benefit from the Shadow Lord (Dragon being one), meanwhile the Dwimmerlaik shuts down so many threats to the Dragon and lets you play without needing Might.

Personally, I would tune up the army a little bit and go with the Dwimmerlaik.

So, which one is better a) 1 named ringwraith b) 2 budget ringwraith? Just asking in general.

there's no real answer to that. A budget ringwraith is a somewhat cheap spellcaster with good magic that can be boosted to fit into your army needs. a Named wraith comes with a special rule that defines its role. You lack a killer? Khamul or the Knight of Umbar will do it. You want to focus on spellcasting? The Undying is the one.
A budget wraith can even be an excellent support to a named one with worst magic.