I am networking to change jobs. I want to train a replacement at my current job. I have identified the person. I need to have more one on one time with that coworker, but it is challenging to arrange.

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.

Having them well trained will also minimize any stress to my employer, though they may not appreciate it, if they knew my plan now. How can I give my coworker the training in the time I have left?

I'm concerned if I propose training someone, the company would never select the individual I have in mind. They like to create paths for their 'favorites'. I want to disrupt that unfair practice by creating a formidable replacement that cannot be argued against and is not going to perpetuate partiality. But perhaps I'm becoming the thing I hate, but being partial myself? Just realized my folly.

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.

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.

"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.

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.

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.