This is caused by a series of problems within the system that is best solved by ret-con and houserule
I’m completely serious: D&D pretends that all ability scores are equal, but they aren’t. Constitution should be almost everyone’s second-highest ability score, particularly low-HD classes that really only ever have need for one ability score (like Wizards with Int). His Constitution shouldn’t just be “not low,” it should actually be quite good. His HP will still be fairly low despite this, because of the d4 HD. I don’t think there’s a lot to be gained from having such low Con that you do not feel you can safely take part in major parts of the game.
Furthermore, Level Adjustment is a terrible system; Wizards apparently actively disliked the idea of players playing as monsters, and quite intentionally over-LA’d most monsters to discourage players from choosing them, or to punish players who ignored that discouragement. I strongly recommend houseruling an LA +0 version of the template, potentially with other drawbacks instead of LA. Level Adjustment, as you have noticed, leads to heavily skewed characters, where they have, e.g., inappropriately high ability scores, but equally-inappropriately few HD (i.e. lower HP, base saves, skills, and so on). This is very bad for the game. The entire LA system is weak. If you indicate which template he has, I’d be happy to give my thoughts on what would maintain the important parts of the template for character/backstory purposes, without requiring the LA.
If you really must have something by the book...
Level Adjustment Buy-off is a start
Unearthed Arcana’s LA buy-off rules are harsh, but with only LA +1 they can work. These work best if your DM handles XP by the book, i.e. “XP is a river” and lower-level characters get more XP so they eventually “catch up.”
I don’t really like this solution because it’s a “suffer now to be overpowered later” which is bad design, and because I despise mixed-level parties: they cause huge headaches for the DM. Also, because this is only a partial solution, since the Level Adjustment is not the only problem here.
Faeries Mysteries Initiate would solve a lot of the problem, but...
The Faerie Mysteries Initiate feat from Dragon vol. 319, among other things, allows a character to use Intelligence instead of Constitution to determine HP. This is grossly overpowered and I do not recommend allowing it. It’s also Fearûn-specific, and race-specific as well, though those things can easily be waived if you like.
If this does get allowed, make it very clear that there will be significant roleplaying requirements, and this is going to be a unique benefit for this character, not a feat that people can start adding as part of their backstory.
Some kind of cursed-but-with-positive-side-effect item?
You could have the character be “cursed” or equip a “cursed” item which gives them more HP. Presumably this is attached to some actual curse that is negative, but maybe it’s an interesting, roleplaying negative that the player won’t mind nearly as much as the character. This fixes the mathematics issue, and adds plot-hooks to boot. But it’s also fairly fiat-y, plus I don’t really have any ideas off the top of my head what the curse would be.
Player’s Handbook II has retraining rules
But they don’t really help here. Just adding this so you don’t waste time digging out the book to check if they’re useful.