Mi hija se quedó mirando el mar desde el banco.

Because the verb here is quedarse, not just quedar.

• quedarse often means to stay, to remain, or to end up in a state
• quedar without se often has other meanings, such as to be left, to suit, to arrange to meet, etc.

In this sentence, se quedó mirando is a very common structure meaning something like:

• she stayed looking
• she remained watching
• more naturally in English: she stood/sat gazing

So the se is part of the verb quedarse and is necessary here.

This is a very common Spanish pattern:

quedarse + gerund

It means that someone remains doing something, or ends up doing something for a while.

So:

• se quedó mirando = she stayed looking
• natural English: she kept looking, she remained gazing, she sat there looking

It often gives the idea that the person’s attention stayed fixed on something.

Other examples:

• Se quedó callado = He stayed silent
• Se quedó pensando = She remained deep in thought / She started thinking and stayed that way
• Se quedaron hablando = They stayed talking

Because Spanish is expressing an ongoing action after a change of state.

• se quedó mirando = she stayed looking / gazing
• miró would simply mean she looked

Compare:

• Mi hija miró el mar = My daughter looked at the sea
• Mi hija se quedó mirando el mar = My daughter stayed looking at the sea / kept gazing at the sea

So mirando is a gerund, and after quedarse it helps show duration or continuation.

Se quedó is the preterite of quedarse.

The preterite is used because the sentence presents the action as a completed event: at some point, she stayed / settled into looking at the sea.

If you said se quedaba, that would be the imperfect, which usually gives a background, habitual, or unfinished feel.

Compare:

• Se quedó mirando el mar = one completed moment in the story
• Se quedaba mirando el mar cada tarde = she used to gaze at the sea every afternoon

So the preterite works well when telling a specific event in a narrative.

In Spanish, nouns often need an article where English does not.

So mirar el mar is normal Spanish, just as you would say:

• ver la televisión
• cerrar la puerta
• abrir la ventana

Even though English might say look at the sea, Spanish naturally uses el with mar here.

Also, after verbs like mirar, ver, contemplar, Spanish usually uses the definite article with a specific thing being observed:

• mirar el cielo
• ver la montaña
• contemplar el paisaje

Both are possible, but they mean slightly different things.

• en el banco = on the bench
• desde el banco = from the bench

Desde focuses on the viewpoint or position from which she was looking.

So:

• desde el banco emphasizes the place from which she was watching the sea
• en el banco emphasizes simply where she was

In many contexts, either could work, but desde el banco highlights the bench as the observation point.

Yes, banco can mean both bank and bench.

Common meanings include:

• bank = financial institution
• bench = seat, often in a park or public place

The context tells you which one is meant. Since the sentence mentions looking at the sea, bench is clearly the intended meaning here.

If it were about a financial bank, the context would be very different, for example:

• Fue al banco a sacar dinero = He went to the bank to withdraw money

Yes, mi hija is completely normal.

In Spanish, possessives like mi, tu, su, nuestro, etc. usually replace the article.

So you say:

• mi hija
• mi madre
• tu hermano

not:

• la mi hija (not standard modern Spanish)

So Mi hija is exactly what you would expect.

Yes, you could, but the meaning changes.

• Mi hija estaba mirando el mar desde el banco = My daughter was looking at the sea from the bench
• Mi hija se quedó mirando el mar desde el banco = My daughter stayed gazing at the sea from the bench / ended up just looking at the sea

The difference is:

• estaba mirando describes an action in progress
• se quedó mirando suggests a moment when she settled into that action and remained there

So se quedó mirando is often more expressive and narrative.

Yes, mirar el mar is perfectly natural.

It is simple and idiomatic. Depending on style, you could also hear:

• ver el mar = to see the sea
• contemplar el mar = to contemplate/gaze at the sea
• observar el mar = to observe the sea

But mirar el mar is the most neutral and everyday option.

In this sentence, mirando el mar sounds very natural in Spain.

The structure is:

Mi hija | se quedó | mirando el mar | desde el banco

So:

• Mi hija = subject
• se quedó = main verb
• mirando el mar = gerund phrase explaining what she stayed doing
• desde el banco = prepositional phrase showing the place/viewpoint

Spanish word order is fairly flexible, but this order is very natural and neutral.

For example, you could also move the last part for emphasis:

• Mi hija se quedó desde el banco mirando el mar

But the original version sounds smoother and more standard.