Thanx for the comments guys.
I actually was more worried about the plural than the "ing" in this case. In both cases though-- like most of you agree-- I believe it depends on the particular word and how it works as all or part of a name.
In many cases I wouldn't want an "ing". In a word such as let's say "hosting", I don't think it hurts, other than giving it three more characters to type and an extra syllable to say. In this particular case, apart from that, "lodging" is better than "lodge". "Budget lodge" didn't give a result on the Overture tool, but just the single word "lodge" produces 9667 results and just the word "lodging" produces 86,972 results. That doesn't mean of course that "budget lodges" or "budget lodgings" will produce many type-ins, but is useful for comparison.
The possible problem for me was the "s", since "lodging" is so often used for the plural as well. To compensate for this "lodgings" suggests a wide choice of places to stay, not just one discount chain, so I still wanted the name despite the "s". Also lodging or lodgings suggests hotels, motels, inns, b&bs, hostels, etc. in one term.
As an aside I noticed it now is taking two or three tries each time to get the Overture tool to work. I wonder if people are running bots or something. It IS Saturday night, we should be out dancing. Or in my case, name-dancing