Is there a preferred type for an Observable with no need for a value in Next events?

You can simply use an Observable<Void> for this purpose. Like so:

   let triggerObservable = Observable<Void>.just()

    triggerObservable.subscribeNext() {
        debugPrint("received notification!")
    }.addDisposableTo(disposeBag)

or in your example:

let triggeringObservable: Observable<Void> 

// ...

triggeringObservable.map { Void -> String in // The actual value is ignored
   return SomeLibrary.username() 
}

In Swift 4 (and probably earlier) you can pass an empty tuple as an argument to a method expecting a Void associated type.

var triggerObservable = PublishSubject<Void>()
...
triggerObservable.onNext(()) // "()" empty tuple is accepted for a Void argument

Or you can define an extension to overload onNext() with no args for the Void case:

extension ObserverType where E == Void {
    public func onNext() {
        onNext(())
    }
}

...
triggerObservable.onNext()

This worked for me on Swift 4.1:

Observable<Void>.just(Void())