Would it be possible to make an alert radio... for shortwave? Out of something like Arduino, or RPi. Sort of like the NOAA Weather Radios. I got to thinking one day. Say you wanted to send a message to a far away friend, but you don't know if he's home or not. Well, what if you could fix that, with a radio that alerts you of the incoming message, and records it as well? I don't think it's impossible, but I've already looked for help, and I've gotten nowhere, so I figured I'd come here. Thanks!
I get the impression that this answer is too long for many people, so here's the abstract: I presume your goal is to make a system that will automatically record a message when someone calls a particular operator on HF SSB and the operator doesn't respond, like an answering machine. Also the system will alert the operator to an incoming call for him or her. Assuming you are talking about SSB, then you'd need some sort of digital message, a selective calling signal containing the call signs of the two stations, that is quickly sent before the SSB call on the same channel. Well there is an existing protocol called Automatic Link Establishment (ALE) that was created for the US military back in the late 70s and early 80s, which is now being used in ham radio, that is used to automatically find the best band and clearest frequency for an SSB QSO. That protocol uses a simple digital mode sent in a short burst before an SSB QSO on the same channel. You could study how ALE works, and possibly find an open-source ALE program and use its code as a starting point, and do some reverse engineering, to create a digital mode and the software to do what you want.
You didn't think this would be easy, did you?
You didn't say what mode you had in mind, but I'll presume that you meant SSB. If you're content with a text-based message, then there is a digital message mode already in widespread use with servers to store and forward messages: Winlink. You did mention short wave, which I'll interpret as 160m – 10m. By the way, I do have an HT that does have a feature that lights an LED when someone calls using a particular CTCSS tone or CDCSS (Continuous Digital-Coded Squelch System) code, IIRC. I think the intent was to let you know someone had called on the calling frequency if you had walked away from the radio temporarily. But that feature works using FM, which allows the subtones in the audio that CTCSS and CDCSS require. SSB doesn't allow such subtones.
To let a radio know that a call is coming for a particular operator, I would think that some sort of digital transmission containing the "to" and "from" call signs would be required. For SSB, I would think that this would take the form of a digital burst before the traditional analog hail. In order to stop the recording, presumably some sort of digital notification of the end of the transmission would also be required.
There is a similar protocol already in use: Automatic Link Establishment (ALE). ALE was created for the US military to allow operators who are unfamiliar with the vagaries of the ionosphere to make SSB contact over HF. The initiating operator gives the ALE controller (a computer, possibly built into the radio) the ALE address of the station to be called. The controller sends a short digital message, a selective calling signal, over many channels on many bands, using a digital 8ary Frequency Shift Keying or 8FSK modulation that uses eight orthogonal tones between 750 and 2500 Hz. The receiving operator's ALE controller listens for calls on the same channels. When the receiving controller picks up a call for it, it sends a message back to the originating station and together the two controllers select the clearest channel on the best band, and then the operators are alerted to make the SSB contact. When the contact is over, one station sends a disconnect signal, and both controllers return to scanning for incoming calls.
By the way, ALE is used in ham radio, but there are disadvantages. As I understand (and I haven't used ALE), in the US, because signals to establish an ALE contact are sent on many channels without direct guidance from an operator, the full ALE protocol suite is only allowed in the narrow sub-bands for automatically controlled data stations. (This page seems to disagree; it says the selective calling features can be used on any SSB frequency. I'm not so sure that allowing a computer to send a digital transmission on any channel that seems to be idle for a few milliseconds is a good idea.) The radio must be able to automatically change frequency and band quickly, so the antennas must have low SWR on all the frequencies used for ALE, or the radio must have a fast automatic antenna tuner.
Anyway, the ALE 8FSK digital mode is intended to encode the call sign of the sending and receiving stations and be transmitted quickly immediately before and after an SSB contact. This sounds very much like what you're asking for, so perhaps you could have a look at ALE and develop a new similar protocol that does what you want.
You might consider looking at packet radio. It allows for this kind of message passing to reach recipients similar to BBS-style messages or email. Similarly winlink is worth checking out.
Neither of these are exactly the point-to-point alert-style you mention above, but both are implementable on a RPi which can be scripted to have a klaxon or spinny light activate when a message comes in.
Hope this is helpful!