What to do with one yen coins when traveling in Japan?

I don't think it would be polite to give a bunch of them to cashier

Foreigners always overthink "politeness" issues in Japan... Handling payments, in whatever (legal) form, is their job, they will dutifully do it without any issues. Just try not to do it during busy times, out of consideration for other customers waiting behind you. If you want to donate them, many businesses such as your nearest McDonald's have donation boxes for various causes, this will be very much appreciated.

This is a problem I frequently find myself with as well. One thing I do to reduce the number of 1 yen coins I have left at the end of my trip is to use them whenever possible. One way I make this easier is at the end of each day sorting the 1 yen coins into a separate part of my coin purse or wallet. When I go to a store and have to pay for something, it's easier to take the 1 yen coins out when they're sorted. I typically use them in situations where I need less than ten 1 yen coins as it typically doesn't take too long to count out that many, given that I've sorted them previously.

They're also good in situations where you would prefer larger change, like if an amount is 2003 yen but you're paying with a 5000 yen note. In this case, I typically put down 3 yen to get an even amount in the end. I'm not sure if this is something people typically do in Japan, but no cashier has ever refused this when I've done it.

Of course, donation boxes are also a great option if you find yourself unable to use them.

I find one-yen coins to be extremely helpful -- but not as coinage, but for electronics repair.

Because they are soft aluminum but also on the thick side, you can use the ¥1 coin to open the "coin battery slots" on many toys and devices. They are soft enough that they won't damage metal or even most plastics and furthermore, they don't have any serrations that would also damage the coin slot. If you bend the coin, you're out ¥1.

So I always keep a couple after a trip to Japan.

The best way to deal with one-yen coins is to never receive them in the first place! All you need to do is two things:

1. Top up your Pasmo/Suica rechargeable smart card and use it at convenience stores, fast food restaurants etc.
2. At grocery stores, larger restaurants etc, pay by credit card.

This will eliminate 95% of cases where you'd get one-yen coins in change, since most cash-only places (small restaurants, attraction entry fees etc) don't want to deal with one-yen coins either and round up their prices to the nearest 10 or even 100 yen.

I don't think it would be polite to give a bunch of them to cashier.

It is impolite to use many coins at the casher. In fact, you can only use up to 20 same coins; otherwise the clerk has the right to refuse it. That said, most people don't know about this law nor would not like to get involved in a redundant argument, so they would start to count and accept it anyway.

But as long as you use up to 20 same coins (even 1 yen coin), it is fine.

Arguably the best way to avoid having a lot of 1 yen coins is to use them more aggressively. Theoretically, you should have up to four 1 yen coins in your wallet as long as you make your payment more carefully.

If you still find too many coins left in your wallet, you might put them in a charity box in front of a casher in a convenience store.

Also note that you must not throw them away on the road or even in a trash bin; this is a violation against the law as well, though no one bothers to report it to police.

In some towns outside of Tokyo you can use them in the buses. Look for those buses where you get a numbered ticket when you come in and deposit money into the machine based on your ticket number when you come out. Most of those don't care for the type of coin you use, so you can dump all your one yen coins there. Just make sure you count them beforehand so that you have the right amount for the fare.

Edit: Since it seems that some wouldn't believe this, here is a quote from Kyushu Sanko bus company website:

?Q.???1???????????????????????

?A.1??5?????????????????????????????????????????????????????20????????????????

Q. Can I use 1 yen and 5 yen coins to pay the fare?

A. We can handle metallic currency that includes 1 and 5 yen coins. However, please aim at using no more than 20 coins at once, since it may cause the fare box to get stuck.

Source: http://www.kyusanko.co.jp/g_others/qa.php

Similar source for Entetsu Bus:

????????????1???5????????????????

"Is it possible to pay the bus fare with 1 and 5 yen coins?"

????????????

"Yes / We humbly accept your payment in such way"

Source: http://entetsubus.lekumo.biz/faq/2006/11/a4_15_e1ac.html

My experience is that most local buses that have that kind of system are able to process one and five yen coins. I guess that it is because they function on an honour basis. That is, the machine is not actually counting the coins, so it can take anything.