A surprising amount of debate that's happened since I first suggested this idea has been about the name. The name used on this blog is just the phrase I used to describe the idea, 'cause it also works as a name, and I felt like I needed to put something as the title.
Some people wanted something amusing or quirky - a name giving the initials "DLA", for instance, or "PIP". "Disability Enforcement Agency" was also suggested, though I suspect more in jest than seriously. Personally, I think serious is better than quirky, but I can still see the amusement in these ideas.
A more serious concern was about the word "union" - that it may create too much of an association with the trade union movement, and might thus be off-putting to conservatives (small c) - and even more so to Conservatives (big C). One self-described conservative, and Conservative voter, objected to that generalisation, and felt that it was the right word.
Another suggestion was to say directly, right in the name, what the organisation is for - Equality for Disabled People was suggested, and I turned it around and suggested Disabled People for Equality, emphasising that the organisation is intended to be by and for disabled people. I'm not sure, though, if that's a good summary when the idea is to be intentionally broad in scope.
I feel that anything we do ultimately comes down to equality, that's for sure, but I'm not sure it speaks to all those meanings for most people. Access to employment, transport, providers of goods and services, that's all equality to most people's understandings (excepting disagreement about vocabulary - more on which below). But would most people put the kind of social security system that works for us under the heading off equality? Those of us who think in terms of a disability equality sort of approach would, mostly, but I'm not sure many people think that way. Of course, it could serve a good purpose in promoting that sort of thinking.
Even the word equality can cause problems - some feel it too closely evokes the idea of equal (by which is meant essentially identical) treatment, and prefer 'equity' or 'fairness' - though I doubt the person on the Clapham Omnibus would understand those the way such people intended, either.
Why is a name so important to people in these discussions? I'm not sure. I think it's important that a name make a statement, and say what you stand for, what you do. I also think it's important not to give a wrong impression, or to confuse people. I don't know if people towards the 'right' (rather than left) end of the spectrum will feel uncomfortable with a union. I don't know if talking about equality in the name will confuse people.
So I have no answers here. I'd love to hear your thoughts in the comments.