We have already heared a little bit of this custom in the MINI encyclopedia, so here are some more theories about this problem.

One easy iconographical feature to discern a Daruma are his big eyes. In the MINI encyclopedia we have already heared that he cut off his eyelids while meditating in front of a wall, because he thought that would prevent him from falling asleep. These big eyes of Daruma are his trade-mark, so to speak.
A Daruma without eyes or rather pupils (menashi Daruma 目無しだるま) is also called "Daruma to make a wish" (gankake Daruma 願掛けだるま). It is usually a tumbler doll made out of papermachee according to the local traditions that are still alive now.
You buy a Daruma without pupils during the New Year celebrations and paint the left eye of Daruma while making your wish. Then you put him up at the Buddhist or Shinto altar in your home. At the end of the year, when the wish has come true, you paint the right eye of Daruma, while giving thanks, then carry Daruma to a temple to have it burned in a holy fire and buy a new one for the next year, and so on and so on.

According to Mr. Kido, the selling of Darumd dolls with eyes started around 1764. During that period, many children suffered of smallpox, which is especially dangerous for the eyes. A Daruma was then used at a talisman to protect from this eye affliction. Since a Daruma with no eyes painted has no special facial expression, the dealers soon sold Daruma dolls with no pupils painted and urged the customers to paint one pupil first and the second after they got better. This custom may have started around 1772. But with the vaccination against smallpox in the beginning of the Meiji period the use of eyeless Daruma as protector for the eyes also disappeared, or rather it changed to other departments of good luck in life.

The story of Daruma dolls is also closely linked to the production of silk and raising silkworms. During the first casting off the skin of the silkworm in spring (harugo), the fist (left) eye of Daruma is painted with the wish that they have many good moltings and grow big and fat. When the silkworms start spinning their cocoons in autumn (akigo) the second eye is painted.

In Takasaki at the famous Daruma temple Shoorin-zan 少林山 you have the eyes painted by the priest. In this case it is the left eye of Daruma. Most other temples follow this rule. 　　

だるまへの目入れ

In Ogaki town, Gifu prefecture there is also the custom to start with the right eye of Daruma, since the right side generally is considered of higher rank than the left. Some politicians also start with the right eye of Daruma.

眼入れのやり方
最初の座禅だるまは、両眼とも見開いていました。養蚕農家が七転八起にあやかって蚕の起き（目覚め「4度脱皮すること」）がよくなるよう大当りの祈願をするため、眼を描かず願いを込めて片目（向かって右）に墨を入れました。やがて蚕が良い繭を作ると、残った片目にも墨を入れて大当りと喜び、お祝いしたのが始まりです。 それが一般に広まって、達磨大師の不屈の精神にあやかり、目標（願い）を立て、精進努力して無事達成するよう願かけをするようになりました。

http://www.city.takasaki.gunma.jp/ad/adb/adb06/adb60000.htm
http://www.wind.co.jp/sanrin/yokuwakaru1.html
http://www.miyagawa.com/syuha/7106.html

願いごとを決めたら、まず向かって右に黒眼を入れてください。そして、願いごとが達成されたら残りの眼に黒眼を入れます。なお、一般 的な事例として、家内安全、健康祈願などに、年の始めに片目に墨を（願いをかける）いれ、平穏無事に1年間過ごせたら、感謝の気持ちで片方の眼に墨を入れる人が多いようです。
http://www.city.takasaki.gunma.jp/ad/adb/adb06/adb60000.htm

There are of course other explanations.
The Buddhist or Shinto home altar common even now in a household is usually facing South. If you want to put your Daruma for Good Luck on this shelf to pray to it during the year, you put up Daruma with the back to the north and paint the first eye, facing east, to start the day and the New Year. Then Daruma can watch over you during the day/year and in the evening the second eye (facing west) is painted. That makes a lot of sense to me. The equation of space and time as being one is very well represented in the statues of the 12 Heavenly Generals (juuni shinshoo 十二神将 ), which each represent 2 hours of the day and one of the 12 regions of the compass at the same time.
だるまを飾る方位にも関係するようで南向きが最適といわれています、玉座（王様や殿様の座る場所）は南に向いているそうです、太陽は東からでて西に沈むので南に向って東は左側になり左目（向って右目）から入れるのだそうです（少林寺和尚様談）。 昔から方位は大切だと考えられ、高崎の少林山だるま寺も江戸時代に前橋城の裏鬼門除けとして建立されたそうです。
http://www.wind.co.jp/sanrin/yokuwakaru.html
http://www.takasakicci.or.jp/present/present1.htm

　　　　　　　　　　　　　　　　　　
If you sit back for a while and meditate about time, usually the idea of the past comes up from behind and your left, while the future is projected into the space before you and to your right. In the temple Jindai-ji the eyes of Daruma are painted in the form of the first and last syllabel of the Sanskrit alphabet, representing the Beginning and End, the course of time in the eyes of Daruma. Of course if you meditate longer, you come to the point to realize that there is NO past and future, but only this very moment, but that is a different problem altogether. Maybe that is another lesson we can learn from a Daruma without eyes. By the way, many Buddha statues are depicted with eyes half closed (hangan) so as to see through time and space, past and future, here and there and all the dualistic concepts we build up in this world.

時間の現象を瞑想の時に浮かぶと過去がなんとなく後ろや左から浮かんで来て、未来のほうが前と左のほうに感じます。深大寺ではだるまの目に梵字の阿吽を書きますがそれも物事の始めと最後、つまり時間の表現です。もっと瞑想すれば時間そのものがなくなるのですが、それはまた別 問題です。目無しのだるまからそれを学ぶことが出来るかも知れません。仏像のほとんどが半眼で作られています。時間や空間、過去と未来、すべてを見通 すように出来ています。

Some people also believe, if you paint both eyes for Daruma at the same time, he has more power to watch over you and make your wish come true. But as one priest explained to me: "You can make a wish and put up a Daruma, but you yourself have to work towards the fulfillment and make a big effort, otherwise nothing will change in your life! Daruma can only remind you at this every day you pray to him." If your wish is too unrealistic to come true, even Daruma cannot help you, so be careful about the things to wish.

　　　　　　　　　　　　　　　　　　　
http://www.optic.or.jp/shop/gift/gangu/shohin/jpg/daruma1.jpg

But how long should we wait until the wish is fulfilled? Until the next New Year? After the Election? After getting well from a disease? Usually the New Year is the time to bring the Daruma back to the temple or shirne, but some people prefer to keep him as a memento and put him up at a shelf. Some carry him back after the election is lost. So basically you can do as you please.

願い事が成就した暁にはもう片方を塗りつぶすんだと思いますが、どのくらい置いておくものなのでしょうか？次のお正月すぎまで大事にとっておくのでしょうか？

その返事はこちらのHPにあります。
http://www.miyagawa.com/syuha/7106-2.html

Here is one more story to explain why, and you find the Japanese in the book by Mr. Minegishi.
In the Genroku period the priest Shunkai Zenji of the temple Kokubun-ji in Nagano prefecture was suffering from an eye disease. He had a statue from a carver in Kyoto of a Daruma with no eyes, so he prayed to this statue for healing. Well, what do you know, his eyes got better and soon after he was completely healed he performed a ceremony to paint the eyes for his Daruma (kaigan shiki 開眼式). The people who heared this story started praying to Daruma figures with no eyes, made of local clay, and soon the temple was famous for healing eye diseases. People who were healed brought their Daruma to the temple to be stored in a special hall.

But let us be franc, does it really matter? As long as the person who gets the Daruma is happy with the result, he can paint the first eye in his favorite direction. What is important is to tell Daruma while you paint: "If you work hard for me and my wish comes true, I will paint your second eye." But there are also folks who paint both eyes at the same time, telling Daruma "Well, I give you both eyes now, so you can work even better for me!" What matters is the sincerity and intensity of your wish, not the location of the eyes.
"Paint the eye as it pleases you most to make your wish come true and work yourself hard to make it happen!" is maybe the best advise we can give to people who turn to a Daruma for good luck.
In Japanese there is a saying "to have good luck" "Me ga deta" (the eyes come out, meaning to have the higher number in a game of dice), or a play with words like "Congratulatory" “ME-DE-tai” (eyes coming out), so the eyes are important symbols for winning good luck. There are some Daruma figures, especially little talismans you buy at a temple or shrine, where the eyes pop out to invoke this saying. Here is one you can even order online at the shrine Tenman-gu in Kobe.

目が出るとか目出たいとか、目に関する縁起の言葉があるのですが、それにちなんで、目が本当に出るだるまさんのお守りや根付もあります。
神戸綱敷天満宮で注文ができます。
http://www.bekkoame.ne.jp/~kunogi/2.htm

A Belgian company about emotional intelligence, time management and self-discipline uses a Daruma for its advertising campaign. FROM a Daruma with one eye TO a Daruma with both eyes,
The Daruma doll from Japan is a visual aid to help achieve results. The pupils have to be added to the whites of the eyes one at the time. The first one when a goal is set, the second one on its achievement. The idea is that the unseeing eye will remind you that the goal has not been reached and so prompt you to take action.
The Daruma doll is used in the Time Manager to remind you of your goals - thus to:
Translate your thoughts into action!
あるベルギーの会社は決断力や時間マネージメントの基本を教えるために目入れのだるまを使っています。片目を見るたびに、自分の力で目的を目指すアクションを起こさせるのダルマ先生です。国際だるまの時代ですね。
http://users.skynet.be/lti/page21.html

This is the first EYE for Daruma in the new millennium 2000.
新世紀2000年の初目入れの風景。

　　　　　　　　　　　　　　　　　　　
http://www.takasakicci.or.jp/present/present1.htm

On the following HPs, you can look for yourself at some people, mostly polititians, painting an eye for Daruma. This is just an arbitrary sample.
目入れのHPはこちらにあります。

　　　　　　　　　　　　　　　　　　　　
http://www.human.tsukuba.ac.jp/lab/mhlab/daruma.html
http://www.idemitsu.co.jp/kankyo/index_t02.html
http://www.root.or.jp/koike/senkyo.htm
http://www.marimo.or.jp/~ebina/dougisen.htm
http://www.sunfield.ne.jp/~s-akio/htm/kouenkai.htm

When making a Buddha statue, the most important part are the eyes, which are usually done last and sometimes a big celebration is connected with this event. Maybe the biggest of them all was the great ceremony held for the Big Buddha at the Todai-ji in Nara in the year 752, where the Indian priest Bodaisenna performed the rites of painting the eyes and music and delegations from all over Buddhist Asia were present.

天平勝宝４年（752年）、大仏の目に筆で瞳を描いて魂を迎え入れる儀式−「大仏開眼供養会」−が行われました。開眼の導師を勤めたのはインドの僧侶、波羅門僧正・菩提僊那（ぼだいせんな）です。
http://www1.sphere.ne.jp/naracity/j/daibutu/w_buddha05.html

Read more about this ceremony and the music on this HP in English.
http://sound.jp/tengaku/Shichseikai-e/shomyo-e3.html

There is a full description of the ceremonies going on at Todai-ji starting on October 15th 2002.
東大寺大仏開眼１２５０年慶讃大法要　日程予定（平成14年）
http://www.todaiji.org/index/1250/1250-10.htm

仏像や仏画、塔婆やお墓の石塔、位牌など私たちがそこに仏や亡き人、祖先を念じ、手を合わせて拝むものすべてに、ほとけのこころを宿すための供養を開眼供養といいます。「み魂入れ」とか「お性根入れ」などというのも開眼供養のことを指します。
http://www.buzan.or.jp/butsuji/qa1.htm
http://www2s.biglobe.ne.jp/~gotosan/226227.htm

Here you find a lot of pictures of such a ceremony for the statue of a Kannon Bosatsu.
越後３３観音札所巡りの第２３番目の札所となっている観音寺の観音菩薩の開眼法要。
http://www.interq.or.jp/joetsu/joshuen/kannonji.htm

Talking about the eyes of Buddha statues, the Great Eyes of some stupas in Nepal come to mind. The eyes of the stupa of Swayambunath are overlooking the town of Katmandu, whereas the stupa of Bodnath is situated in a natural mandala in the middle of the valley and the huge eyes seem to see anything that is going on in the area.
仏像の目の話なら、ネパールの佛塔の大きな目も忘れてはいけない。最後にこの素晴らしい目を見てください。

　　　　　　　　　　　　　　　　　　
http://www.sacredsites.com/1st30/swayambh.html

　　　　　　　　　　　　　　　　
http://www.klausdierks.com/Himalaya_Deutsch/spuren_der_lamas.htm
The problem of Japanese religiosity and the Daruma cult will be studied in different stories.
達磨さんと日本人の宗教生活は別の話になります。