Here’s an analysis of whether the relative pronoun can be omitted in each sentence:

1. **This was the hotel that we stayed in for two weeks.**
- **Omission:** Yes, you can omit "that."
- Revised: This was the hotel we stayed in for two weeks.

2. **She showed me the emeralds which she had brought back from South Africa.**
- **Omission:** Yes, you can omit "which."
- Revised: She showed me the emeralds she had brought back from South Africa.

3. **A doctor is a person who looks after people's health.**
- **Omission:** No, "who" cannot be omitted here, as it is necessary for clarity.
- Revised: A doctor is a person who looks after people's health. (remains the same)

4. **My new motorbike, which I paid a few thousand euros for, is not running well.**
- **Omission:** No, "which" cannot be omitted in this non-defining clause because it adds essential information.
- Revised: My new motorbike, which I paid a few thousand euros for, is not running well. (remains the same)

5. **Where is the letter that arrived this morning?**
- **Omission:** Yes, "that" can be omitted, but it may affect the formality.
- Revised: Where is the letter arrived this morning? (more informal and less common)

In summary:
1. Omit
2. Omit
3. Do not omit
4. Do not omit
5. Omit (informally)