Programming

You can probably use Audacity to generate a tone. The more you can find out about the specs of the original speaker, the easier it’ll be to generate a new tone.

Actually, resulting volume amplitude from pc speaker can be tweaked by using pulse-width modulation (PWM) techniques. It does not work that well with cheap moving-iron pc speaker but still possible to play PCM waveforms with this trick. Some DOS software like games such Pinball Fantasies or MOD players like Inertia Player supported this.

Most classic 1-bit pc speaker info just has codified frequency and time on and off programmatically, so it is suggested to find non-pcm and unlikely varying, non-standardized ways to store this information. 1-bit does not have some specific shape, unless you imagine a square waveform but actually on an analog signal it is far for being real square due to vibration, so you get the Gibbs phenomenon represented by Fourier series primitives (all sounds are just a combination of sine wave harmonics). But you don't throw those primitives to the digital controller, it is just the natural analog resulting effect of sound from digital information.

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If you have the hardware, you could record the beep by removing the PC speaker and wiring those terminals directly into an audio recorder or the mic jack on another computer. That way you'd have the exact sound you want.

Understandably not everyone can do this, so an alternative could be to record the sound with a normal microphone and then use software to determine the frequency of the beep so you can recreate it in Audacity or something. I don't know the names of any off by heart, but I'm sure there will be some free ones online.