As you have noted, there is not much in the way of official material to compare this item against, which makes getting an accurate feel for the item's power by theory alone difficult. However, I would put this item somewhere around the upper threshold of rare, which might be a bit stronger than you want (although take this rating with a grain of salt).
To gauge the power level of such an item, increasing the effective level of all necromancy spells by one, we would have to investigate what increasing all necromancy spells by one level would do. Below I have compiled a list of all necromancy spells in the Player's Handbook which explicitly benefit from being cast at higher levels, sorted by the original spell level.
- False Life, +5 temporary hit points
- Inflict Wounds, +1d10 damage
- Ray of Sickness, +1d8 damage
- Blindness/Deafness, +1 target
- Animate Dead, +2 creatures
- Vampiric Touch, +1d6 damage (and heal for half)
- Bestow Curse, duration 10 minutes (higher at higher levels)
- Blight, +1d8 damage
- Circle of Death, +2d6 damage
- Create Undead, +1 creature (extra options at higher levels)
I note that some of the powerful damaging necromancy spells, namely harm and finger of death, do not benefit from being up-cast.
For necromancy spells, the damage bonus which would be provided by your magic rod is not that high. Consider that a +1 weapon is an uncommon magic item and a +2 weapon is a rare magic item. +X weapons always add +X to damage and to hit. Your rod provides a boost to damage of between about 4 and 6 for some certain spells. Note that it already costs you spell slots to cast spells, so this is not a continuous buff like a +X weapon, although you probably have more spell slots than most items have daily charges. If damage alone were the only consideration, and noting how there are only a small number of damaging necromancy spells, this magic rod would probably be rare. Compare this to (for example) the mace of disruption, mace of smiting or vicious weapon, which provide comparable situational bonuses to damage.
Of course, damage is not the only consideration. There are various non-damage increases. The most significant are bestow curse and create undead. By using an item to cast these spells at a higher level than you could normally cast, you gain access to options with these spells which you would not normally have access to at your level (different creature types for create undead; avoiding concentration for bestow curse). This is an important consideration.
But there is another effect of casting at higher levels besides explicit 'At Higher Levels' effects. Some counters to magic, such as dispel magic, counterspell, globe of invulnerability and a Rakshasa's Limited Magic Immunity, are dependent on the level of the spell being cast on them. This rod would make it harder for your necromancy spells to be dispelled or countered, and would also circumvent many things which block spells of 5th level or lower once you have access to 5th level spell slots (rather than when you gain 6th level spell slots).
Because of the combined increase in potency and difficulty in countering, I would posit that this magic rod sits somewhere between a rare and very rare item. The Dungeon Master's Guide recommends rare items for characters of 5th level or higher, and very rare items for characters of 11th level or higher. 11th level is when full spellcasters gain 6th level spell slots and thus become able to circumvent globe of invulnerability and a Rakshasa's Limited Magic Immunity.
I could theory-craft all I like, but the only way to really know how powerful this item is is to test it out. It is possible that the various edge cases where being able to cast at a higher level than normal could be unbalanced are rare enough not to be of any concern. Or it is possible that by specialising in appropriate spells that this item could be used to much more devastating effect than I anticipated.
If you are worried that it might be too powerful, then putting a limited number of daily charges on it is a good idea. Many of the magic items in the Dungeon Master's Guide have a limited number of charges, and those which don't have limited charges for a higher level effect tend to be rarer.
You can tell your player that this item is homebrew and subject to being redesigned. If you decide that the limited daily charges are not needed after testing, you can remove the charges and make its use unlimited. Making the item a little too weak then buffing it later is something your player would like better than making the item too strong then nerfing it.