What is the origin and basis of the idea of spending an hour each day in prayer? In my society there is this odd thing being discussed or requested of people to pray i.e to spend one hour a day praying.
I do not understand why one needs to pray for one hour.  The closest is the time Jesus asked if his disciples can not stay watch for one hour but this does not translate to one hour pray.
Can any one explain the notion of one hour prayer and where in the Bible does it teach us how long to pray?
When asked to teach his disciples how to pray, Jesus responds by saying 'when ye pray' but does not specify when, or how often or how long, Luke 11:1.
He then gives an example of prayer (and advises not to use it as mere repetition for he warns against such a thing elsewhere Matthew 6:7) and the example takes but a minute or two to actually voice.
There are many examples of prayer in scripture but I have never yet found a rule or a requirement about time or frequency.

And it came to pass, that, as he was praying in a certain place, when he ceased, one of his disciples said unto him, Lord, teach us to pray, as John also taught his disciples. And he said unto them, When ye pray, say, Our Father which art in heaven, Hallowed be thy name. Thy kingdom come. Thy will be done, as in heaven, so in earth. Give us day by day our daily bread. And forgive us our sins; for we also forgive every one that is indebted to us. And lead us not into temptation; but deliver us from evil. [Luke 11:1-4 KJV]

Daniel (for example) used to pray three times a day but this was only discovered when he was put under constant surveillance so that some fault might be discovered whereby he could be prosecuted as prayer was forbidden at the time to any but the king, see Daniel 6:10-28.
But Daniel never commands anybody else to pray three times a day. It was just what he chose to do. Similar examples may be found in the psalms where psalmists promise (to God) to pray or give thanks. But this is between them and God himself. They do not demand it of others.
In the garden of Gethsemane, Jesus was in an extremity. He wanted company and support. But the disciples were weary and could not stay awake to watch during the hours of the night.
In the end God, the Father, sent an angel to strengthen Jesus in answer to his prayer, see Luke 22:43.