As jstimpfle mentioned, that doesn't work, which is why the argument is left off the alias. In general, you can do that in a function instead of an alias:

    lsr() {
        find . -name "$1"
    }

Wrapping variable expansion with double quotes is a good habit, so spaces are handled properly.

Also, if you are using a modern-ish bash, you almost always want to use "$@" when there could be multiple arguments. The double-quoted @ special variable is guaranteed to always expand as multiple args, but with spaces handled correctly:

    foo() {
        bar --quux=42 "$@"
    }

    foo "a b c" "Spaces in my filename.txt"

That doesn't work the way you think. Aliases don't receive arguments. The $1 will be expanded to the first positional argument of the surrounding environment when you call the alias -- not the first argument of the alias invocation.