Does ASL have language games? They seem very common for spoken languages (e.g., Pig Latin for English), but I can't find any documentation of them for sign languages.
Deaf humor is primarily visually based. A scholarly article on the subject by Swarthmore College is summarized as such:
Sign language humor in America and Britain may be seen in the creation of new visual signs, the witty reanalysis of existing signs and in bilingual games in which English is manipulated within sign languages.
A page on LifePrint outlines a couple of deaf jokes, of particular relevance to your question being the following:
Production and misproduction of signs is a common way to elicit laughs in ASL. One example, described in Bellugi and Klima's book, The Signs of Language is how we can change the root sign, UNDERSTAND, to LITTLE UNDERSTAND by using the pinkie rather than the index finger.
This same author also lists a few ASL idioms on a different page*:
- TRAIN GONE = missed opportunity to know what is being talked about
- CIGARETTE GONE = missed opportunity to know what is being talked about. Note, this is a clever twist on the "train gone" idiom since some people say, TRAIN BACK! But you can't bring back a smoked cigarette.
- FINISH TOUCH = been there, have physically been to a place
- FISH = "I am done. It is over." This is a pun / idiom based on the fact that many Deaf when doing the sign for "FINISH" make a mouth movement that looks as if they were saying the word "fish."
- BASEMENT = Stayed home, didn't go out.
- "BY-A-HAIR" (pull a hair) = "Whew! That was a close one!" The signer does a sign that depicts the "pulling of a single hair" but the actual meaning has nothing to do with the pulling of a single piece of hair.
- FISH-SWALLOW = gullible. The signer does a sign which depicts the swallowing of a fish but the actual meaning is that someone is gullible.
- BLOW-BRAINS-OUT = Gee, oh wow, I can't believe that it (a certain piece of information) isn't coming to my mind. I know this but can't think of the right word, or information.
- SCRATCH-FOREHEAD = I will never forget that. / "Scarred for life."
- STRICT = "hard nosed" = unyielding, not flexible. This sign is interesting because the meaning of the sign is interpreted as "strict." You see the sign and you think "strict" -- you don't think of it as being an idiom. But if you consider the likely history of the sign you can see that it is a combination of the signs "HARD" and "NOSE." The phrase, "He (or she) is hard nosed," is obviously an idiom since the literal meaning has nothing to do with having a "hard nose."
*The author indicates not to copy/paste this list without permission... given this is from a 2005 email hosted on the page, and I cannot find anywhere to contact Dr. Bill (author) to ask permission, I am going to assume linking and acknowledging this request is sufficient.
I realize this isn't quite the direct type of language game you were looking for, but it is certainly a type of "play on words" within ASL, so I thought it could be helpful.