What's the difference between at hand, on hand and in hand? At hand seems to me as if you have something in reach. On hand is if you have something in stock. And in hand can be used as if you have more time at your disposal.
The way these different prepositions are used is all very much idiomatic, so there will be a limit to how far you can "rationalise" why any particular preposition is associated with any particular meaning. Especially in respect of "hand", since most of us always have a couple of those about our person ready to be incorporated into our word choices.
One thing I would say that "in" usually denotes something closer than "on" (which itself is usually closer than "at"), which is why the "nearby" meanings use the latter two.
Note that "in hand" doesn't actually mean having more time at your disposal - it's just that we often say we have "time in hand" (things in hand can be in reserve, spare). Apart from the literal held in the hand, the other common idiomatic usage for this one is that if something is "in hand" it's being actively dealt with (by implication, really close to you), not just sitting on your "to-do" list.
OP doesn't mention to hand, which is actually more common than the others in British English (but seemingly quite rare for Americans). In practice few would choose at hand or on hand on the basis of how physically close/accessible something was - largely pointless anyway, since few others would apprehend any subtle distinction being made. Go British and make life easier for everyone!
At hand frequently refers to proximity in time, the other two usually relate to spatial proximity:
With the apocalypse at hand, I was pleased to find I have a cache of food and water on hand, and a rifle in hand.
Think of measures of distance:
in hand - literally within one's grasp - not even inches away - "A bird in the hand"
at hand - nearby - a few feet away at most - "He kept his cigarettes at hand"
on hand - somewhere accessible but fairly distant - more than a few feet away and less than a few hundred yards - "We have more vacuum cleaners on hand"
I tried to break down the usage of these three (four if you count "to hand"), hopefully I'm right about it, at least in part :) So here are my thoughts:
Examples: 1) Pupils have over half an hour in hand = left/available/to spare. 2) We had our kitchen redone and still have some savings in hand = still left (didn't spend all of them). 3) The troops entered the village to take the situation in hand = to take it under control/to reign it in. 4) The matter in hand should be smoothed over with senior managers = the matter that is now being settled/the present issue that worries somebody.
Examples: 1) The businessman was afraid that retaliation was at hand = would befall him soon/would soon follow. 2) The stranded group of tourists knew that help was at hand = a rescue operation to save them was commenced, and the rescue team would soon get the tourists off the island, they knew that =). 3) You might want to have a pen close at hand/to hand = have it nearby, somewhere close by so that it is easily accessible if you need it right away.
Examples: 1) The emergency services were on hand with medical advice = they were available/they could be reached (over the phone, for instance) for some important medical advice. 2) He's often on hand to help you out = feel free to ask him if you're puzzled about something/he can often be of help if you ask him about something. 3) The company needs to have cash on hand = it has to have cash available for some purposes, the cash should be easily accessed, so to speak. 4) Are there enough people on hand to get started? = are there enough people present (maybe they are members of some committee or board) to hold a meeting? 5) A change was on hand (=at hand) = it was imminent/about to happen soon.
While there are rules there is also common usage, it's worth remembering that these prepositions can occaionally be interchangeable. It's advisable to refer to dictionaries and pay attention to the way native speakers use these prepositions, TV shows can come in handy for that matter as well. =)