From Wizards' July 2015 Sage Advice column:
Does the warlock’s Awakened Mind feature allow two-way telepathic communication? The feature is intended to provide one-way communication. The warlock can use the feature to speak telepathically to a creature, but the feature doesn’t give that creature the ability to telepathically reply. In contrast, the telepathy ability that some monsters have (MM, 9) does make two-way communication possible.
Agreed that there are too many powers and abilities here, and they're too far out of scope for a familiar. Let's see..
blindsight: the only base familiar with blindsight is the bat, and his is echolocation. So, not inherently broken, but rare.
burrow speed: Only familiar with a burrow speed, and tries to make up for it by tanking speed in general. This one is also... a bit odd. Why would an eldritch crystal have a burrow speed? More to the point, it lets you get in and out of certain places rather more easily than you'd otherwise be able to at low level, and gives it a near-guaranteed escape if it's on a surface it can tunnel in. Would be acceptable as a Unique And Shiny Thing, or convert it into a flight speed (for levitation). Possibly remove the ground speed entirely and give it only a flight speed (probably a slightly faster one) to represent the fact that it's a crystal and does not actually have legs. Lets it be a bit weirder (I suspect the player likes weird), but not in a way you need to worry about.
otherworldly resistance: This is... excessive. Its primary effect is that it lets the player throw around area effects a much as they like without hurting their familiar. It's also not well-justified. In fact, as far as I'm aware, nothing in the game has blanket immunity to spell damage. If he were building a completely different familiar, I might accept immunity to a single element as the one completely awesome thing it had, but that's not what he's building here. Carve it out.
innate spellcasting: Blur is second level, and increases survivability in a way that other familiars can't match. Meld into Stone is third, and offers both survivability and shenanigans. Both are overpowered, compared to what is expected of familiars. The only other familiar with a spellcasting effect is the octopus, who's water-bound.
intelligence stat: 10 is way too high for a familiar base. all other base familiars have between 1 and 3.
constitution stat: higher than any of the base familiars in the book (most of whom have 8). By extension, this is the only familiar with 3 hp. Probably pretty harmless in itself, but underlines the min-maxxing.
In general, then, the familiar as presented has far more survivability than any of the base familiars, as well as being weirdly intelligent.
Suggestion: As a base, keep the blindsight. Convert the burrow and land speeds into a moderate fly speed (levitate). Let him have the con. Drag the base int down to the 1-3 range (possibly with a special feature to let him use the arcana skill with a non-negative modifier). Ditch the otherworldly resistance and the innate spellcasting. At that point, you're a few notches down from a bat (significantly slower flight speed, no land speed, and no attack. The Arcana thing is cute, but not a big deal, and the other advantages are relatively meaningless). You can probably fit in one more interesting thing without breaking the bank. Possible "needs not eat/sleep/breathe" and make his slowish (10 or 15) fly speed also be a swim speed. That lets him explore (slowly) into underwater areas without having to be aquatic, and lets his master cast spells based on noxious fumes at areas that contain him without concern. In the end, it wouldn't be doing anything that other familiars couldn't do, but the combination would be rare and useful, and it would be doing them in a different way, which should appeal to the player involved.
edit: your player has expressed a strong desire for fire immunity. That's a specialized but situationally very powerful effect, which his own tendencies are going to make somewhat less situational. If you want that, it should be pretty costly. Given the character in question, I'd suggest you lose the "needs not eat/breathe/sleep" and swim, lose the blindsight, make the familiar a permanent light source and possibly give it a familiar-esque 1 damage melee attack (fire). Call it a Forge Spark - a sort of least fire elemental associated with the character's god.
Best Answer
It depends on the familiar.
The default find familiar forms are all beasts, and list no languages spoken. Nothing in the find familiar spell alters that; it says you can communicate with it telepathically, but nothing mentions giving it the power of speech, so it doesn't.
However, some of the bonus forms for the Warlock's Pact of the Chain do speak. The Sprite, Imp, and Quasit speak Common, among other languages, and could act as a translator in the way you describe.