From the Manual of the Planes, p153:
Githyanki Silver Weapon
Power (At-Will - Psychic): Free Action. All damage dealt by this weapon is psychic damage...
That seems pretty straightforward.
From the Manual of the Planes, p144:
Primordial Blade (16th level): ... At the end of each short rest or extended rest, select one of the following damage types... Your attacks deal 1d4 extra damage of the selected damage type until the end of your next rest.
The key here is the use of the phrase "extra damage", which has a very specific meaning.
From the Rules Compendium, p223 (emphasis mine):
Many powers and other effects grant the ability to deal extra damage. Extra damage is always in addition to other damage and is of the same type or types as that damage, unless otherwise noted...
So extra damage isn't the same type as the damage it's added to if a type is specified for the extra damage (which it is, in this case).
This really comes down to whether you think the rule for extra damage trumps the silver sword's conversion of all damage to psychic or vice versa. Specific beats general, so powers beat general rules if they say they do; the question is whether the silver weapon's rather vague "all damage dealt by this weapon" is meant to include extra damage. Is Primordial Blade's damage even being dealt by the weapon?
Neither game element received any errata while 4e was being actively supported, and there don't seem to be any other rules that would clarify this case, so ultimately you'll probably have to ask your DM to rule on the matter.