As Brian Goetz pointed out in a comment, the basis for the design decision was to allow adapting a method to a functional interface the same way you can call the method, i.e. you can call every value returning method and ignore the returned value.
When it comes to lambda expressions, things get a bit more complicated. There are two forms of lambda expressions, (args) -> expression and (args) -> { statements* }.
Whether the second form is void compatible, depends on the question whether no code path attempts to return a value, e.g. () -> { return ""; } is not void compatible, but expression compatible, whereas () -> {} or () -> { return; } are void compatible. Note that () -> { for(;;); } and () -> { throw new RuntimeException(); } are both, void compatible and value compatible, as they don’t complete normally and there’s no return statement.
The form (arg) -> expression is value compatible if the expression evaluates to a value. But there are also expressions, which are statements at the same time. These expressions may have a side effect and therefore can be written as stand-alone statement for producing the side effect only, ignoring the produced result. Similarly, the form (arg) -> expression can be void compatible, if the expression is also a statement.
An expression of the form s -> s can’t be void compatible as s is not a statement, i.e. you can’t write s -> { s; } either. On the other hand s -> s.toString() can be void compatible, because method invocations are statements. Similarly, s -> i++ can be void compatible as increments can be used as a statement, so s -> { i++; } is valid too. Of course, i has to be a field for this to work, not a local variable.
The Java Language Specification §14.8. Expression Statements lists all expressions which may be used as statements. Besides the already mentioned method invocations and increment/ decrement operators, it names assignments and class instance creation expressions, so s -> foo=s and s -> new WhatEver(s) are void compatible too.
As a side note, the form (arg) -> methodReturningVoid(arg) is the only expression form that is not value compatible.