You could do either, but I think you will have to factor in periodic growth in database for either case. During the expansion of datafiles database will be slow/unresponsive. (There might be a setting so this happens in the background - I forget ).
A related question - MongoDB performance with growing data structure, specifically the "Padding Factor"
With first approach, there is an upper limit to number of websites you can store imposed by max number of collections. You can do the calculations based on http://docs.mongodb.org/manual/reference/limits/.
In second approach, while #of collection don't matter as much, but growth of database is something you will want to consider.
One approach is to initialize it with empty data, so it takes lasts longer before expanding.
For instance.
{
website: name,
responses: [{
time: Jan 1, 2013, 0:1, ...
},
{
time: Jan 1, 2013, 0:2, ...
}
... and so for each minute/interval you expect.
]
}
The downside is, it might take you longer to initialize but you will have to worry about this later.
Either ways, it is a cost you will have to pay. The only question is when? Now? or later?
Consider reading their usecases, particularly - http://docs.mongodb.org/manual/use-cases/hierarchical-aggregation/