Surely many of "those who have come to Britain to make Britain their home" (especially where that has been a free choice), have done so with an appreciation of "British Values"?
Even if we don't like the phrase itself. The underlying meaning of it - a combination of identity, ways of doing things and behaving, ways of organising our society etc is a reality.
Surely many people who settle here from other countries do so because they feel they would benefit from those things. They are a big factor in making Britain what it is. And we may not be perfect but we're not that bad - else people wouldn't come here.
Surely then they are to be celebrated? Why shouldn't we have a cultural identity? And in doing so we aren't saying that no other countries have any of the individual 'values' just that this mix is the British mix.
What exactly it is, is a harder question to answer, and maybe it can't be done, though I think many people on here have done a good job.
And all the arguments on here against having one do seem to fit with the idea that part of our cultural identity is self-deprecation!
I'd go for....
Equality of opportunity, tolerance and humour, combined with 'just getting on with it'.
I love that line someone said about how we tolerate stuff until it 'just gets silly' (so British) then we say no.