My suggestion? Don't.
When I run sandbox games, I tend to divide the world into regions of general power; I start the players off in a low threat zone, full of mudcrabs, rattata, and the occasional goblin. Then, in universe, I tell the players what areas are safe. Rumors in the bar that the road to Harborhead has been having some bandit troubles. The city watch warning them as they leave town to take the left fork, not the right one, because the Ghostgate is full of terrifying spirits. A traveler who explains he and his whole town had to pick up and leave because a dragon took up residence in the mountains to the north. These hints allow the players to decide how much they're willing to risk; the loot and XP for taking out that dragon will be incredible, but it's very likely to roast them alive on the first round.
It sounds like you've played 4th edition in the past, which had a fairly strict power curve. 5th edition actually works even if the characters aren't at the same level as their opponents, helped by the fact that there isn't as big a difference between low and high level characters. Still, you and your players may have some habits to break.
As players, they need to learn how to pick their battles, estimating which fights they can win and setting things up to give them a leg up. They also need to learn when to retreat and back off. Some fights aren't winnable, for reasons nothing to do with levels. As a DM, you need to learn to foreshadow encounter difficulty clearly. (Honestly if you feel like you haven't made something clear, it's okay to drop out of character for a moment. "You can totally fight the dragon today if you want, but you guys should know I set him up to be a level twenty battle and I'm going to play him that way. If you beat him though, his hoard and XP will be on the same scale.")
5th edition plays to a slightly different tune than 4th, but for a sandbox game like yours, it should be an easier one to dance to.
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
Rules as Written - Stay the MM
Familiars, Rules as Written, don't advance. The Pact of Chain Familiar doesn't improve stats over time. The only text we have on them is in Find Familiar and in the Pact of Chain class feature.
Pact of Chain
Find Familiar
Upgrade with Magic
A sprite is powerful in stealth, has poison arrows, and great for flavor in a feylock. Unfortunately, it isn't very hearty. You can improve its armor (and weapons), or give it with temporary health points by magical means. The sprite familiar takes its existing equipment with it when sent to the pocket dimension, so in theory, the upgraded equipment should work go as well (there is a question on this, but even if the sprite couldn't take them with them, they are light and you can carry it). Magic items meant for PCs (isn't true of some of the weapons monsters carry) resize to the size of user, so there is no reason it couldn't use magic weapons/armor.
Ask your DM
Any more than this would require you get your DM agree to house rule for progression.