When you cast a spell...
The problematic phrase in the description of Overchannel is "When you cast a wizard spell of 5th level or lower that deals damage" because 'when you cast a spell' can have different meanings in English, and these multiple meanings are supported at different points in the rules.
'When you cast' could mean 'At the single point in time, and only at the point in time, in which you cast a spell...' Typically this meaning is used for features that involve a choice or selection made specifically at the time of casting.
For example, from Tiny Hut,
Creatures and objects within the dome when you cast this spell can move through it freely.
and
The evocation wizard's earlier Sculpt Spells feature says,
When you cast an evocation spell that affects other creatures that you can see, you can choose a number of them equal to 1 + the spell’s level.
Conclusion: 'when you cast' can refer to just the moment of casting, for decisions about the spell made at that point in time.
On the other hand, 'when you cast' could mean, 'If you cast a spell, the following consequences occur while the spell is in effect'. Typically this meaning is used for features that have ongoing effects for the duration of the spell.
For example, just about every spell description that includes additional effects when the spell is upcast begins:
At Higher Levels. When you cast this spell using a spell slot of nth level or higher...
and the description of the effects often do not apply only at the time of casting, but later as well. For example, in Wall of Fire,
When you cast this spell using a spell slot of 5th level or higher, the damage increases by 1d8 for each slot level above 4th
and this increase in damage applies for every round the spell is in existence, up to its full duration.
Conclusion: 'when you cast' can also refer to the general situation of you casting a spell, and can introduce a condition that applies for the entire duration of the spell.
Thus, in order to evaluate how Overchannel works, we need to determine what kind of 'when you cast' it is using. Does it apply to damage done only at the time of the spell casting itself, or does it apply to all subsequent damage done by the spell?
Note that this concern is not limited to just the OP's example of Arcane Hand - it equally applies to any spell that can cause damage over multiple turns. Even though most Evocation spells are like fireball in that they cause damage only once, on the round when they are cast, there are many evocations that have the potential to cause damage over multiple rounds, such as Witch Bolt, Acid Arrow, Fire Shield, Wall of Fire, and Cloudkill. The cases of Fire Shield and Cloudkill are especially illustrative, for they never cause damage simply 'when cast', but only subsequently. Restricting Overchannel to damage caused only by the spell casting itself would make it useless for spells like Fire Shield and Cloudkill.
Furthermore, since Overchannel applies to any "wizard spell of 5th level or lower that deals damage", an Evoker can also use it on their conjuration spells, where the list of multi-round damage spells is much longer.
Thus we need to think carefully about the effects of a ruling here - if we say that Overchannel maximizes damage done only by the act of casting the spell itself, then we are drastically affecting its ability to maximize damage for many evocation spells. Alternatively, we can see that allowing Arcane Hand to have maximized damage each round really does not set it apart as a special exploit, since many other evocation spells are in the same category.
If permitted, just how powerful is this?
Suppose we Overchannel a fifth level fireball as our standard for an evoker. A rolled 10d6 has an average damage of 35, but a maximized damage of 60, for a difference of 25 in one round (assuming a failed save, but applicable to multiple targets).
Compare this to an Overchanneled Arcane Hand. Used for a Clenched Fist, the rolled damage is 4d8 per round. Here the average is 18 and the maximum 32, for a gain of 14 points (less than the gain in fireball). But this is just one round. Over ten rounds, that gain could be a difference of 140 points, which does seem like a lot at first blush. But we must discount this by a quarter, for when the attacks don't hit (assuming +10 to hit and AC of 16), so it is down to 105. Then discount it for the likelihood that the fight will be over long before ten rounds; in a typical three-round fight, the total expected gain from Overchannel is now 35 points - which is less than the gain to an Overchanneled fireball if we allow the fireball to have two targets that fail their saves. We could discount the value to Overchanneling an Arcane Hand further for the difficulty of maintaining concentration on the spell, the opportunity cost of maintaining concentration on the spell, and the fact that the Evoker's Empowered Evocation can be used once per spell only, but if a different spell was cast each round it could add damage multiple times.
We end up in an interesting decision space. Overchanneling a single direct-damage spell like fireball is the better option if you want to do more damage in one round. Overchanneling a damage-over-time spell like Arcane Hand can certainly result in a greater damage gain per spell slot if you can and will use the same spell for more than three rounds. Perhaps a rough rule of thumb would be to Overchannel the damage-over-time spell when the number of rounds you expect the spell to last is greater than the number of targets you can get in the area of your competing direct-damage spell.
That is a tactical tradeoff that results in having to make interesting choices, not a broken exploit that makes you select Arcane Hand every time.
RAI
Interestingly, Jeremy Crawford has tweeted that Overchannel was intended to work "the first turn" only - but that RAW it "lasts as long as the spell lasts". And specifically with respect to Arcane Hand, he doesn't consider that a problem.
