Instantaneous effects cannot be continuous by the definitions of continuous and instantaneous. If he actually got what he wanted, it wouldn’t do what he expects.
The duration “Instantaneous” means that the magic changes something about reality in a real, permanent way. It does not have a duration because the magic is gone, the change already done. For HP damaging and healing effects, the change to the HP is permanent, even though the magic is fleeting. This is a very important property.
Compare, for example, wall of fire and wall of stone: the latter makes an actual wall. It’s there, it’s made of stone, there’s no magic to it – anymore. The magic just made it. You cannot dispel it, it does not show under detect magic, it remains even if you bring an antimagic field to it. It is now a permanent part of reality. That is what Instantaneous means. By contrast, the wall of fire is an ongoing magical effect: you can dispel it, and eventually, its magic will run out and it will disappear.
Now imagine, for example, if cure light wounds had a Permanent duration instead (comparable to the Continuous duration). This would mean that the magic remains active on the target, which has a number of effects, none of which are equivalent to being healed 1d8+CL HP per round as your player seems to expect. Rather, you would be healed by 1d8+CL HP at the beginning of the effect, but then would lose that HP if the spell were taken away. If, for example, it was dispelled. Moreover, if you had cure light wounds cast on you again while the first was still running, the two would not stack, since they are the same source of the healing. Instead, you would just be healed by however much more (if any) the new casting had healed you.
So your player really doesn’t want a Continuous item of cure light wounds; it just doesn’t do any good, and arguably could interfere with actual healing he might get (if it came in the form of additional cure light wounds castings). This is probably true for just about every Instantaneous effect out there: changing the duration has problematic, nonsensical interactions with the rules, and in (probably) every case, makes the effect much, much less valuable. Instantaneous is, by quite a lot, the best duration possible.
The conclusion here is that you really can’t make Continuous or Permanent versions of Instantaneous effects. Not necessarily because the rules say you can’t, and really not even because it wouldn’t be useful (though it wouldn’t be), but because it really just doesn’t make sense. While I’ve “imagined” a Permanent version of cure light wounds above, in reality there are a number of issues with such a thing that I have just ignored for the purpose of the imagining, but would cause problems if you ever tried to actually do it. Thus, rather than simply being a bad idea, I would argue it’s actually impossible because the rules just do not define what would happen when you do it. Instantaneous effects are designed to be so and don’t make any sense having another duration.