I have a question regarding present perfect and simple past

I have this example:

Have you ever lost something valuable?

The answer is : No, I haven't, but my brother ___ his camera on a trip once.

My question is this. With what should I fill the space? When should I use present perfect and when should I use simple past? Do I write lost or lose? I am confused between them.

You fill the space with lost because the past tense describes an action (or event or process) that occurred at a particular time in the past. The words 'on a trip once' identify just such a time.

"That would be "... my brother lost his camera...".

Present Perfect has four functions:

• An event that began in the past and is still continuing ~ "He has worked here for 10 years", where he started to work there in 2002 and still does.

• An event in the past with present results ~ "They have developed some nice MP3 players", where the the MP3 players existing now is the consequence of past development.

• The 'pre-present' ~ "He has just arrived", where he arrived a minute ago

• The 'indefinite past' ~ "Malaria has killed millions of people", where the time malaria started killing is unknown, but in the past.

Your brother once losing his camera is none of these, and is definitely over ("...once...") so it would be a simple past.

It should be: "But my brother lost his camera on a trip once."

The present perfect is never used when you're referring to a specific time in the past (even if you're only being as specific as "once"). It is used to indicate an action that has taken place at some unspecified time.

Consider: "The workmen have completed [present perfect] their task," versus, "The workmen completed [past simple] their task on Wednesday."

While all the answers so far are correct, the OP's question seems to have two spaces, thus implying the need to use a two-word tense construction.

There are two compound past tense forms that are acceptable in this context. The more fitting one is an alternate form of the past simple:

No, I haven't, but my brother did lose his camera on a trip once.

This form uses a participle of the actual verb (almost always formed by taking the infinitive minus the word "to"), plus the past simple tense of "do" (which is always "did" regardless of subject) to form the past simple tense. In the general case, this form is synonymous with the conjugated past simple ("lost"), but it typically has the connotation of either contradicting a prior negative (in this case "no I haven't"), and/or emphasizing that the event indeed happened. I can find no online reference defining this construction as anything other than past simple tense, and oddly I can't find any reference that uses the "do" verb to construct this tense in the affirmative (only the negative), but it definitely exists in modern speech.

The other possibility, which is less fitting, is the present perfect:

No, I haven't, but my brother has lost his camera on a trip once.

This is marginal; while it would work to express the fact that your brother, at one point in the indefinite past, lost a camera, the word "once" feels redundant; the use of the present perfect coupled with "once" makes it seem like the count of the number of times it's happened (once) is more important to the speaker than the event (losing a camera). If the sentence instead ended with "many times", this tense would be the only valid answer, but as it is it feels a little clunky.