What is the best night to pray qiyaam if you can only do that once each month?

There is no particular night which Islam tells us to single out from the other nights of the week or the month, rather you can pray whenever it is easy for you to do so and when you feel energetic. If you can pray qiyaam on more than one night each week or each month, that is better for you than praying only one night. 

You should avoid singling out the night before Jumu’ah for praying qiyaam, because of the clear prohibition on doing so. 

It was narrated from Abu Hurayrah (may Allah be pleased with him) that the Prophet (peace and blessings of Allah be upon him) said: “Do not single out the night of Friday from other nights for praying qiyaam, and do not single out Friday from other days for fasting, unless it is part of a fast that one of you regularly observes.” Narrated by Muslim, 1144. 

Al-Nawawi said: 

This hadeeth clearly indicates that it is forbidden to single out the night of Friday from other nights for praying, and to single out that day for fasting. It is unanimously agreed (among the scholars) that this is makrooh. 

Sharh Muslim, 8/20. 

But if a person prays qiyaam on Friday night, not because he is singling it out because it is Friday, but because it is a day when he is off work, then there is nothing wrong with that. Similarly if he fasts on a Friday because it is a day when he is off work, there is nothing wrong with that. 

Shaykh Ibn Baaz said: 

If the Day of ‘Arafat happens to fall on a Friday and a Muslim fasts on this day only, there is nothing wrong with that, because he is fasting because it is the Day of ‘Arafat, not because it is a Friday. Similarly, if he has days to make up from Ramadaan and he has no free time except on Friday, then there is no sin on him if he singles it out; that is because it is a day off for him. Similarly if the day of ‘Ashoora’ happens to fall on a Friday, there is no sin on him if he singles it out for fasting, because he is fasting because it is ‘Ashoora’ not because it is Friday. Hence the Prophet (peace and blessings of Allah be upon him) said: “Do not single out Friday for fasting, and do not single out the night of Friday for praying qiyaam.” So this hadeeth refers to singling it out, i.e., when a person does that just because it is Friday or the night of Friday. 

End quote, from Majmoo’ Fataawa Ibn Baaz, 15/414 

Shaykh Ibn ‘Uthaymeen said in al-Sharh al-Mumti’ (6/291): 

It is makrooh to single out Friday for fasting, and the evidence for that is the fact that the Prophet (peace and blessings of Allah be upon him) said: “Do not fast on Friday unless you also fast a day before it or a day after it.” And he said: “Do not single out Friday for fasting or the night of Friday for praying qiyaam.” And he said to one of the Mothers of the Believers, when he found her fasting on a Friday: “Did you fast yesterday?” She said, “No.” He said: “Are you going to fast tomorrow?” She said: “No.” He said: “Then break your fast.” If she had fasted another day with it, he would not have objected. So if you fast Thursday and Friday, there is nothing wrong with it, or if you fast Friday and Saturday, there is nothing wrong with it. 

If a person fasts on Friday, not because he is singling it out, but because that is free time for him, such as a man who works every night of the week and has no free time except on Friday, is it makrooh for him (to fast on a Friday)? 

The answer, in my view, is that it depends. If we look at the report narrated by Muslim: “Do not single out Friday for fasting,” we say that there is nothing wrong with it, because this man is not singling it out as such. But if we look at the hadeeth ‘[He said:] “Did you fast yesterday?” She said, “No.” He said: “Are you going to fast tomorrow?” She said: “No.” He said: “Then break your fast”’, then it may be understood from this that it is makrooh to single out this day. But if he cannot fast on any other day, then it may be understood differently, and it may be said that what the Prophet (peace and blessings of Allah be upon him) said – “Did you fast yesterday… are you going to fast tomorrow?” – indicates that she was able to fast on other days. 

The point is that if a person singles out Friday for fasting, not because it is Friday, but because it is the day when he has free time, then it seems in sha Allah that it is not makrooh, and that there is nothing wrong with that. End quote.