If a psion uses the 8th-level shaper power astral seed [metacreativity] (Expanded Psionics Handbook 78) and stores the resulting seed in the realm created by the psi-like ability personal space, would the psion, upon his death, regrow from his own body?
The psion can never hide in his own personal space
This is stopped by a number of details. First, the personal space is limited to items only: you, a psion, can, by expending your psionic focus, "send an item to [the personal space] or retrieve an item as you wish." Creatures aren't ever referred to as items, and in the game, item is usually synonymous with object. (Ask the DM about intelligent magic items, though, which, like shrubbery, straddle the line.) Second, the personal space description says that in the personal space "[y]ou cannot store anything that has extradimensional properties," and your body has extradimensional properties due to the special ability personal space. (Your body has extradimensional properties all the time—you just need to expend your psionic focus to access these properties.) Finally, the the personal space is "within your body," and you tend to take that with you wherever you go, so no shoving you into yourself.
Ask the DM what happens to items in the personal space when you die
Certainly, the seed from the power astral seed can be put in the personal space, but if the psion dies while it's there, it's up to the DM to determine what happens.
This DM's preference would have the contents of the personal space spill onto the plane the psion's body occupies, as if the psion were a bag of holding (DMG 248) (2,500+ gp; 15+ lbs.) and had been turned inside out. Using this option, for example, the astral seed stored in the space would appear, glowing, near the psion upon the psion's death. This is simply the path of least resistance for playability: it's terrible when, for example, a party member is given the MacGuffin for safekeeping and the MacGuffin ceases to exist.
Because that is another strong option: the psion's death causes everything in the personal space to be lost forever as if the psion's body were a bag of holding that'd been pierced or overloaded. Using this option, for example, the astral seed stored in the space is lost forever—as is, probably, the psion's life force, the astral seed growing on some unique, unnamed demiplane, forcing the psion to employ the 5th-level psion/wilder power psionic plane shift [psychoportation] (XPH 124) or similar means if he wants to see his friends again (or, y'know, suicide upon realizing his fate and hope his friends notice). While this seems the more likely outcome to this reader, this result is so frustrating—for players and the DM—that, really, I'd go with the first.
Finally—and this is likely the option you want—everything stays the same upon the psion's death, all the items happily sitting in the psion's belly dimension (at least I hope it's his belly dimension!), waiting for the psion to be returned to life. The psion is not like a bag of holding, the personal space is, and it continues its existence just fine upon the psion's death: it's just a unique demiplane, after all, and all the psion does is access it using his psionic focus. I mean, in theory, it even exists semi-independently of him: while he can't stuff creatures into himself, a creature capable of employing, for example, psionic plane shift could visit the psion's personal space if he really wanted to and the space permitted. (The psion still can't travel there himself because it's still specifically "within [his] body.")
Using this option, an astral seed could totally grow in a personal space, but once grown, the psion would have to escape from himself, likely leaving him in the same position as the second option—either psionic plane shift or suicide. While present in his old body's personal space—because he's managed to smuggle an extradimensional space into an extradimensional space where there should not be an extradimensional space—this DM would likely rule that his present body's personal space is inaccessible. A less generous DM may, instead, have something terrible happen (this despite there being no consequences for putting a bag of holding inside a bag of holding).
(By the way, when the power astral seed says, "Upon [your life force's] transference, your physical remains (should they still exist) become inert matter and cannot thereafter be restored to life," this reader groks that as a limitation on the corpse and not as any effect-ending side effect of the power astral seed. The term inert is also used in the description of the 8th-level Sor/Wiz spell clone [necro] (PH 210)—which the power astral seed closely resembles—therefore seeming to this reader to be synonymous with soulless (cf. the 5th-level Sor/Wiz stone to flesh [trans] (PH 285) and the 5th-level Sor/Wiz spell flesh to stone [trans] (PH 232) which specifies it turns creatures into inert statues). While being soulless is pretty bad—seriously, it's pretty much the worst part of gaining the condition dead, for instance—, so far as I'm aware, special abilities don't normally cease to exist or even end because a body lacks a soul.)
As I mentioned, because it's easy and playable, this DM would rule that items in the psion's personal space are expelled from the personal space upon the psion's death, but ask your DM what he prefers… and the consequences of that preference.
Personal Space says "You gain an extradimensional storage space within your body." Astral Seed says "Upon transference, your physical remains (should they still exist) become inert matter and cannot thereafter be restored to life." By extension, while you are dead and reforming your seed, you would not have your personal space, as it can no longer be supported by your body. What happens to things inside the space when you die is unclear, but there's a good chance it's not "staying in a perfectly safe zone that you control".
Practically speaking, it's an ask your DM moment - especially because you might care about what happens to the contents of your Personal Space on death well before level 20.
Yes, but only exactly that long
Your personal space cannot contain anything that has extradimensional properties, which your body, fully formed, certainly does. However, an Astral Seed doesn't, and neither does the ectoplasmic proto-body until you regain the appropriate class feature, at which point you have finished regrowing. So, you can hide your astral seed in your personal space, you can possess it when you die, and you can regrow there, but once you are done regrowing the space can no longer store you (what happens then is not specified).
Also, somebody made a wisecrack about breathing in the comments, so I'll go ahead and point out that so long as you are in your personal space you must be a sentient psicrystal, which grants you the Construct type, which removes your need to eat, sleep, or breathe, so the resuscitation will not run into problems for that reason.
What happens to the personal space when you die?
Well, there's a couple things in play here.
1) your Astral Seed, once you have possessed it, renders your body inert matter. 'Inert' is frequently used in 3.5 rules text to indicate than an object has been rendered non-magical, and this does not seem to be an exception. Once the seed is possessed, your body is rendered inert, and e.g. its ability to link to an extradimensional space is lost.
2) What happens to objects in an extradimensional space when that space ceases to exist is unclear. For Bag of Holding-like objects (which your personal space references but doesn't actually say it functions like), "all contents are lost forever", which makes you pretty safe. For spell-like effects, it's complicated but probably you get dumped out. If it's like Leomund's Secret Chest, you're just left wherever you were, but it's not connected anymore.
You should probably ascertain what will happen to the space when you die before committing to this.