Consider the spell levitate (2nd level), which allows the caster to lift a hostile creature off the ground and hold it in place. It is easy to make the comparison to spells like hold person (2nd level) and hold monster (5th level). While the hold spells paralyze the creature, levitate prevents a creature from moving, and from making melee attacks.

My DM suggested that the target of levitate be able to repeat the save every turn, as he felt that levitate was too strong in comparison to hold person and hold monster. Is this balanced?

The hold spells paralyse, levitate does not

The main difference I can see between hold person/monster and levitate is that the former paralyses the target, whereas the latter only lifts the target upwards.

From hold person (PHB, p.251):

The target must make a Wisdom saving throw or be paralyzed for the duration

And hold monster (PHB, p. 251):

The target must make a Wisdom saving throw or be paralyzed for the duration

Compared with levitate (PHB, p. 255):

One creature of your choice that you can see within range rises vertically, up to 20 feet, and remains suspended there for the duration [Concentration, up to 10 minutes] ... An unwilling creature that succeeds on a Constitution saving throw is unaffected.

A creature affected by levitate can still do many things, including making ranged attacks and casting their own spells that require saving throws, whereas someone who is paralysed can do nothing.

From the Paralyzed condition (PHB, p. 291):

• A paralyzed creature is incapacitated (see the condition) and can't move or speak

From the Incapacitated condition (PHB, p. 290):

• An incapacitated creature can't take actions or reactions.

Since a paralysed target is a sitting duck, it seems much more fair to allow them to make a saving throw each round to recover from this, since they have no other options; someone affected by levitate, on the other hand, can still do many things, just nothing that requires melee range or (depending on the room, since you can apparently move yourself around if you can "climb" along the walls/ceiling) movement.

So I'm concluding that, because the effects of hold person/monster are more severe, the target "needs" to make a save each round more so than someone affected by levitate.

Duration

The other consideration, however, is duration. Hold person and hold monster both last a minute, whereas levitate lasts for 10 minutes. Therefore, if someone doesn't really have a lot of options (for example, if a melee-focused character who has, say, run out of javelins) is potentially stuck in the air for 10 minutes, that will take them out of the fight for longer than someone who is paralysed (if they have literally no other options to contribute), although they will at least not be as vulnerable as someone paralysed (i.e. they can take the Dodge action, as Theik points out, imposing disadvantage on attack rolls against them, compared to the advantage and auto-crit imposed by the "paralyzed" condition).

That said, most combat doesn't last more than a few rounds, so in a typical combat, this will never be a concern. I'd still lean on the side of suggesting that allowing a saving throw each round for levitate would weaken levitate.

What about a Strength check rather than a Constitution save?

Another idea could be, rather than following the example of hold person, perhaps borrow from entangle or similar? Being held in the air against your will, whilst it doesn't actually restrain you with regards to the "restrained" condition, does effectively restrain a melee-focused character. Entangle restrains one or more creatures as per the "restrained" condition, and provides a different way to end that condition.

If someone affected by the levitate spell were to be allowed to use their action to make a Strength check to break free of the caster's hold on them, this might be more balanced; they'd still have to burn an action, rather than getting a "free" saving throw, but it does still give a melee-focused character suspended in mid-air an option to do something about it.

For reference, here's entangle (PHB, p. 268):

A creature restrained by the plants can use its action to make a Strength check against your spell DC. On a success, it frees itself.

It will ruin the innate balance of spells

Look at the two examples you've listed, hold person and levitate.

Levitate: When used as a hostile spell, if they fail their initial save, you're fairly capable of holding them there for ten minutes. They can still attack, they can still cast spells, they can still take the dodge action. You can take a melee-only opponent out of the fight, but that is it.

Hold person: You inflict paralyze on the opponent, one of the strongest effects in the game. They auto-fail on certain saves, they can't move, speak or act, any attack against them has advantage against them and any attack from within 5 feet is an automatic critical hit. This is huge; but the downside is that they get to save at the end of every round.

You are suggesting giving the only part where Hold Person is weaker to Levitate as well. If somebody can save against levitate after every round, why would anybody want to cast levitate instead of hold person?

There are certainly spells in D&D that will seem unbalanced, but trying to balance them yourself is almost certainly going to lead to new balance issues.