As said, Dynamic Signal Analyzers are sort of the go-to name for the older equipment designed for this specific use. Since then, digitizers have become cheaper and having a dedicated instrument for just this hasn't really justified a dedicated instrument in that many current lineups. Modern scopes will all give you FFT analysis capability, but to get a reasonable dynamic range compared to a DSA, you'd probably want a 12 bit scope with enhanced resolution from oversampling - an 8 bit converter just doesn't compare.
Otherwise, there are a number of full blown spectrum analyzers that go down to very low frequencies, some end in the kHz band, but many offer low frequency extensions to 100Hz, 20Hz, or 10Hz. This is also somewhat common on older SAs, whereas it seems to be an optional extra on more modern instruments. The TG may not go down to this frequency, but any signal generator would, so by interconnecting the trigger signals and a little on-instrument configuration, you should be able to get a bode plot pretty easily.
Otherwise, the modern equivalent is probably an audio analyzer. Very high dynamic range, low noise, and usually with fast enough converters to manage 100kHz+ worth of bandwidth. Not so inexpensive of an option, but with the internal signal generator to cover the range and give you more comprehensive testing than just a TG.
PC and frontend/generator/software is probably the cheapest to get really good dynamic range, a scope with more than an 8 bit converter and a signal generator is probably the simplest/cheapest option new, and an older DSA is probably not too expensive of an option for the specialized bit of gear. A soundcard and software is a great inexpensive option if you can verify that it doesn't filter out stuff above the audio band - a lot of sound cards will rolloff above 20kHz just to keep the system noise level down.