We can read CR-appropriate as "not changing her damage output or effective HP"
In the absence of an explicit definition, we can just read the passage ordinarily. An alternative interpretation of that passage would go something like: "If you give the hag the ability to cast extra spells, give her spells that don't modify her CR".
So what determines CR? In the DMG tables, CR is the average of the offensive and defensive CRs. The offensive CR is determined by damage per round (assuming that the spells the hag casts are at her spell DC). Therefore, any spell that does damage that's equal to or less than her average damage output would be appropriate. Conversely, any spell that does not significantly increase her effective HP/AC would also work.
For example, giving the hag Power Word Kill would significantly increase the hag's offensive CR, and thus would not be CR-appropriate. However, an item that casts Astral Projection might still be CR-appropriate, since it doesn't deal damage in combat and it doesn't increase her survivability.
This calculation kind of breaks down when you get to spells that remove or incapacitate PCs without actually dealing damage, such as plane shift or forcecage. While the Save DC column of the DMG table partially addresses this issue (the save DC doesn't change based on spell level), it will be up to your judgment as to whether such spells are too strong for a CR3 hag.
Ultimately, the true answer is the least satisfying: you'll have to test it out yourself (DMG 275):
After seeing your monster in action, you might want to adjust the challenge rating up or down based on your experiences.
Identifying spell effects currently affecting an item/creature or spell origins of an item, learning the number of charges of a magic item, and dramatically reducing the time expense.
You can use the identify spell to identify any spell effects an item or creature is currently being affected by or the spell used to create an item (see the spell description from the SRD or PHB).
You learn whether any spells are affecting the item [you touch throughout the casting] and what they are. If the item was created by a spell, you learn which spell created it.
If you instead touch a creature throughout the casting, you learn what spells, if any, are currently affecting it.
The short rest option for identifying the behavior of magic items does not replicate the options above, since the short rest option only applies to magic items (and, depending on how the rules are interpreted, maybe even only those that require attunement).
In addition, you learn how many charges a magic item has.
If [the item you touch throughout the casting] is a magic item or some other magic-imbued object, you learn its properties and how to use them, whether it requires attunement to use, and how many charges it has, if any.
The short rest option makes no mention of learning the number of charges, although it is common for a DM to rule otherwise (see the description of this activity in the Magic Items, Attunement section of the rules from the SRD or DMG).
[A]t the end of the short rest, the creature gains an intuitive understanding of how to activate any magical properties of the item, including any necessary command words.
Finally, the above advantages are in addition to the reduced expense of time. In order to use the short rest option to learn a magic item's properties and also become attuned to it, you actually have to spend 1 short rest learning its properties and then another becoming attuned.
Attuning to an item requires a creature to spend a short rest focused on only that item while being in physical contact with it (this can't be the same short rest used to learn the item's properties).
That's a total of 1 hour of resting to learn some of its properties (or 2 hours if you also want to become attuned), as opposed to a 1 minute casting time to learn all of its properties (or 1 hour and 1 minute if you also want to become attuned). Since identify is a ritual, it can also be cast in 11 minutes without expending a spell slot.
Best Answer
Are there effects that fool the spell Identify, and what are they?
Identify (PHB p.252)
Note that it can be cast upon a creature, which a lot of people don't know as, like I did until I re-read it for this answer, base their knowledge of the spell on older versions that could not.
Effects that may fool an Identify:
Non-detection (PHB p.263)
An Identify spell will return no information on a creature or object with this spell in effect.
It is up to the DM whether the language "cannot be targeted" means that RAW the spell simply fails as it cannot target the object or creature for some reason and thus they may realise that something is up or, in my opinion RAI, it works but the Identify is fooled into thinking there is no magic to target and so returns no information normally as if it was non-magical.
Amulet of Proof Against Detection and Location (DMG p.150)
This has the same power and issues as Non-Detection above, in that it uses the "can't be targeted" language.
Nystul's Magic Aura (PHB p.263)
The wording of the spell means that the Identify is given false information, allowing the caster of the Aura spell to make it appear non-magical if it is magical.
However the reverse is problematic as, RAW, the Aura spell does not provide any false spell effect or magical property information, just the school of magic it radiates. The information provided would therefore be odd and suspect.
Mind Blank (PHB p.259)
Divination simply does not get any information on the target so an Identify will not identify any spells running on the target. The language here does not have the "cannot target" stipulation given in Non-Detection so it does not have the issue noted there.
Divine Intervention (PHB p.59)
It is within the DM's power to identify the cleric's need as somehow including an effect that would fool an Identify spell (which is most definitely within the power of this, as scaled by the "the effect of any cleric spell or cleric domain spell would be appropriate") and choose an effect appropriately. The DM is not constrained to a cleric/cleric domain spell, that is just given as one appropriate possibility.
And of course Wish (PHB p.288) (if you are willing to suffer the consequences
You can duplicate any of the effects above, or you can go for something outside the normal bounds of the spell, e.g. "I Wish that this object appeared in every way to inspection and divination magic as being a flametongue". Though given the power of a Wish and the consequences of casting it this way: "I wish this sword to be immediately and permanently enchanted as a Flametongue" may be a better bet, depending on how evil your DM is! Obviously the second is "more powerful" and therefore more risky...
As a sideways look at this question, it would be possible for the following spell to be used to "fool an Identify spell" in a different manner:
Modify Memory (PHB p.261)
Instead of making the Identify spell get incorrect information, you could change the caster's memory of its results.