As a matter of fact, we know that the yield on zero coupon government bonds of all maturity change over time and that those changes aren't perfectly predictable. As this is a source of risk, it should be compensated and, therefore, the theoretical ideal would be to introduce a stochastic short rate process in your option pricing model.
Some people did just this across a nice set of models: Bakshi, Cao and Chen (1997) looked at the Black-Scholes-Merton model, the Heston model, a jump-diffusion model, as well as a jump-diffusion and stochastic volatility model. Whether you add a stochastic interest rate or not in any of the above, your pricing errors do not diminish and your hedging performance doesn't improve... The reason? Well, think about it for a minute: the small changes in interest rate that occur in a period of a few days to a few months do not command a huge premium. As long as you take into account the fact that in the real world, interest rates have a term structure, you can "cheat" your way out of modeling its behavior.
That's what you'll see academics do in research papers. They'll look for a close match in terms of maturity on a government bond and use the yield at time t on that bond to price all options that have a similar time to maturity at time t. From the stand point of your price process, they will "cheat" by working with empirical excess returns:
\begin{equation}
ln(S_{t+1}) - ln S_t - r_{t,T}.
\end{equation}
Now, there is a problem with this approach, namely that this isn't a very good approximation of the true cost of capital for trading desks and market makers. So, one thing they could do is pick another interest rate. You could opt for the LIBOR rates, US Treasury repo rates or the Sterling Overnight Index Average instead of the US Treasury bond yield curve. As far as I can tell, this seems to be the simplest way to go. You avoid the modeling problem, it's easy to implement and it takes into consideration the fact that rates always show some kind of term structure.
Using a fixed rate seems odd to me. The rates change and they have a term structure.