There are several options, and they are all worth considering.
The various options will all vary in cost, short term, and long term stability.
Short term stability generally included changes in frequency due to temperature, noise in the signal, and short term fluctuations. This would affect readings taken 5 minutes to a day apart.
Long term stability is often dominated by aging of the crystal (for crystal oscillators), temperature drift in ovens (caused by aging of the temperature regulation components), etc. This would affect readings taken a week, a month, or a year apart.
- Keep using your function generator. This is simple, available, and almost no cost. However, it deprives you of the function generator while using the counter, and is possibly limited in accuracy (depending on your function generator).
- Make your own Crystal oscillator. You may even have a suitable module, if not it's a simple and cheap thing to do. However there accuracy won't be really good and it will drift with temperature.
- Buy a TCXO from eBay. This is simply a crystal oscillator using a temperature stable crystal. Chances are, the unit has one of these internally and you may even be able to fit it in place. You get a 10MHz signal that is more stable with temperature.
- Get an OCXO. An ovenised crystal oscillator has an inbuilt heater and temperature control (the oven) used to keep the crystal at a constant temperature. These are even more easily available on eBay, and second hand units are often reasonably priced. Some of these units have a double oven. This means that the crystal is inside one oven, and that oven is inside a second oven. There oven can draw a significant amount of power (I have one that draws in excess of 15W when started from cold). The power requirements drop once they heat up. Beware of OXCO's that don't draw a lot of power, the oven may be broken. Also realise that these may require 30 minutes or more to come up to temperature. I have a counter with one of these fired internally and it is always peered (even when the main switch is in standby -- it doesn't have an off position).
- GPS 10MHz signal. There are a heap of cheap gps modules. Find one with a 10MHz output. Their long term stability is excellent, being similar to that of the hydrogen maser on the satellite. However, they suffer from short term drift, and require the GPS to be able to see at least 3 satellites. Aside from not working in really nasty conditions when GPS signal is lost, and potentially needing an external antenna, this is pretty much set and forget (there is nothing to adjust). It is also becoming very cheap.
- Rubidium oscillator. These are generally units removed from decommissioned cell towers. They have really good short and long term stability. Again, these use a lot of power on startup. I think mine consumes 25W or so. They also have a limited lifetime. These can take up to 10 minutes to come into lock, but they do have an output that tells you they are locked.
- GPS disciplined oscillator. This is a combination of one of the other oscillator types with a GPS to improve the long term stability. Mine has a double oven crystal oscillator. These can take 30 minutes for the oven to warm up, and maybe 10 minutes for the GPS to lock, but it's still not over, because the units need to know their precise position to correct the signal they receive from the satellites. They may need to be powered up for several weeks before they achieve their ultimate accuracy. Mine is designed to run from DC. This is not uncommon because you would normally have it run from batteries so it could remain powered during outages. If there is a GPS loss (really heavy rain can do it) the internal oscillator will continue to give you an output that is as good as it's short term stability until the GPS signal is recaptured.
- Hydrogen maser. Good luck getting one of these
Depending on your needs, 2, 3, 4, or 5 will probably be what will work for you.
Some googling will give you indicative comparisons of short and long term stability of these options.
This doesn't compare everything (and it does list a few more options), but it may be useful.
Oh, I also should mention that a second hand crystal oscillator may actually have very good long term stability because frequency changes caused by aging slow down with time. However, the device may have been removed from service for a reason (you takes your chances).
I compare my GPS disciplined oscillator against the unknown 10MHz source using my oscilloscope. When locked to the GPS source you can watch the other source drift (and adjust it where possible). Once you get them accurate to a fraction of a Hz, you can leave everything running for a few days to monitor how the signal drifts (although you have to make assumptions as to which signal is doing the most drifting).