How can I give my coworker the training in the time I have left?

You don't, this is your boss's job to assign not yours. When you resign from the company, your boss will decide who ( if anyone ) will be trained to fill your role. They will also decide if you will be the one doing the training or not.

Present the training activity to both the trainee and your managers as improving the bus factor. It is prudent to have at least two people capable of doing any job.

Talk to your manager first, so that you are certain who they want you to train.

Training the wrong person could create an extremely difficult situation when you resign. Training person X would make them expect to be your successor, and leave them feeling resentful if your manager prefers person Y. Your X training effort would have been wasted, and you would have to do hurry-up training of Y during your notice period.

"Getting hit by a bus"

You're training someone to do what you do so that if you're out sick one day, get hit by a bus, or have some other emergency your company has someone ready and trained to fill in your role until you come back (or don't).

Its just a level of insurance that your company's management should see as a "good thing" not something to fear that you're intending to leave. It just so happens to let you extricate yourself without leaving the company in the lurch when you do hand in your resignation.

Before all of that surreptitious training the easiest and best way to do this is to write a whole bunch of great documentation and comment your code. That way no matter who gets picked after you're gone will have a good starting position. Even the person that comes after them will benefit (especially if it is kept up to date.) You also won't need to worry so much about training them up yourself, they can learn after you leave.

But if you really want to try to secretly train this specific person you'll need to get them to start working on the system. You could try claiming illness or just being overworked and ask if they could cover just this one "little task" for you. Start getting them to work on little bits of the system. Let them ask you a bunch of questions about it.

This isn't really a morally okay thing to do. Your co-worker isn't going to appreciate it while you're doing it. You may get in trouble for not doing your own work. The only way (except for good doco) that they will learn about the system is to work on it. So you'll need to find some way to get them to do that. (Alternatively just tell your co-worker what is going on and let them study the system and ask you questions. As long as they won't tell your boss.)

My bosses cannot know that I am considering leaving, because they will
make my worklife miserable.

I want my coworker to be well established to fill my role and be in
line for promotion and a good opportunity after my departure. We are
not close, but I know that they are in such a situation as to benefit
greatly personally and professionally. So I also do not want to let
them in on my plan. Especially as it may cause bad repercussions for
them if they are found to knowingly go along with my plan.

So your plan is to sneak around and be deceptive, both to your bosses and your colleague?

That doesn't sound like a good plan.

I can understand your not wanting to tip your hand to your bosses, but involving your colleague in your plan without their knowledge is deceptive, dishonest, and could potentially put their job in jeopardy.

I'd caution you to rethink your approach to this issue.

Don't make plans for other people, assuming you know what's best for them.

This includes your employer and the coworker you wish to train to be your replacement.

You've got good intentions here and that's commendable. But the fact is, your version of how people will behave and what they'll want to do once they learn you're leaving is likely wildly wrong.

Frankly, people usually overestimate the impact of their leaving on the company and their colleagues. This is expected - you're obviously the center of everything that you do at work, and so you see everything as it relates to that - so naturally everything you do looks especially impactful and important. Other people have a different viewpoint.

In this instance, it's very possible that the colleague you want to train to replace you simply isn't interested at all in doing your job, for example. Or even if they are, that management would have no intention of promoting them into the position anyway. What you see as an obvious opportunity, worth twisting yourself in knots to give to this person, may turn out to just be nothing.

If you want to leave, then secure another position first, then do the following:

• Give your contractually agreed notice to your employer. Mention nothing about your colleague at first, but ask them if they're ok with you telling people straight away.


• 
  If yes, then go to your colleague and ask them privately if they'd be interested in your job.


• 
  If yes, suggest to your colleague a handover training plan to best line them up for this. Make sure they know this is you suggesting this, not the employer, and you'll suggest it to the employer if they agree.


• 
  Only if they are OK with it, then go back to your employer, tell them your colleague is interested, sell them as a good candidate, and suggest the handover plan.


• 
  If your employer agrees it's a good idea, implement it.

Assume nothing. Do everything in the open. Make suggestions, sure, but let everyone involved take their own decisions about what will happen at every stage, before you proceed.

Tell your boss you are quitting. They really can't do much to make your life hell. Just work your fixed hours at a regular pace and go home. Make sure to use all your holiday before you leave. As you are the one leaving, you have a lot of power. Your boss will want you to train a replacement. If they treat you badly, you will just leave.

EDIT

Do this once you find a new job, not before.

You don't because the only ethical way to do what you suggest is if you could train this person on your own time, on your own equipment, outside of business offices. Or if your employer has a loop-hole allowing you to spend hours training people however you like which it doesn't sound like.

But perhaps I'm becoming the thing I hate, but being partial myself?

Absolutely. The fact you have your job proves that your employer is not as bad as you paint them. Unless you're saying that you don't really deserve the title/role you have. So we know that they aren't always bad & that they (according to you) aren't always good. That describes like 99% of the employers people actually want to work for, you know.

At any rate it's your employer's prerogative to manage employee positions. Not yours. Sometimes employers ask employees for advice/opinions on promotions but that's not the case here. Maybe if you told your employer that you're leaving they would give you the opportunity to help them train/select someone. Obviously you'd also face the risk of being shown the door before you want.