Star Eyes

In manga and anime, sometimes characters are drawn with "stars in eyes," me ni hoshi 目に星, or their "eyes become stars," me ga hoshi ni naru 目が星になる.

This can happen for various reasons: the character is thinking of something dreamy, yearning for something; that's just part of the design of the character; or, in rare cases, they just took damage somehow and the stars symbolize getting hurt.

Character: Tsukino Usagi 月野うさぎ
Anime: Bishoujo Senshi Sailor Moon, 美少女戦士セーラームーン (Episode 1)

• Examples
  
  • Temporary
    
    • Single Eye
  • Permanent
• Wink
• Four-Pointed

Examples

For reference, some examples:

Temporary

When character's eyes turn into stars, they're probably thinking of something dreamy.

Character: Oginome Ringo 荻野目苹果
Anime: Mawaru Penguindrum, 輪るピングドラム (Episode 4)

Single Eye

A star drawn in a single eye is used when a character is surprised, shocked, in disbelief, or when they're hurt by something.

In western cartoons, the same effect is typically achieved by drawing stars floating over a character's head. This is related to the English phrase "seeing stars," which means seeing flashes of lights when you bump your head, for example.

Character: Delia Ketchum, Hanako ハナコ
Anime: Pokémon, ポケットモンスター (Episode 1)

• Context: Ash Ketchum's mother got electrocuted by his electric rat.
• Hair strands sticking out are another symbol typically used when characters are overwhelmed.

Character: Hoshizora Miyuki 星空みゆき
Anime: Smile Precure!, スマイルプリキュア！ (Episode 39)

• Context: a shoe falls on Miyuki's head.

Character: Asagaya Minoa 阿佐ヶ谷未乃愛
Anime: Animegataris, Animegatarizu (Episode 3)

• Context: Minoa is shocked by an unexpected development at the third episode.

Permanent

Some character designs feature star-shaped pupils which are permanent and most likely don't mean anything.

Character: Rekka Hoshimiya 烈火星宮
Anime: En'en no Shouboutai 炎炎ノ消防隊 (Episode 7)

Wink

Sometimes a star comes out of a strong wink. This doesn't mean anything. It just looks cute.

Character: Mephisto Pheles メフィスト・フェレス
Anime: Ao no Exorcist, 青の祓魔師 (Episode 17)

Four-Pointed

In anime, four-pointed stars are sparkles, and if they're in a character's eyes, those are sparkling eyes.

Character: Kise Yayoi 黄瀬やよい
Anime: Smile Precure!, スマイルプリキュア！ (Episode 35)