Is there any science (or trick) to determining a replacement op amp?
Start with power supply requirements - does the prospective replacement work on the power rails supplied? Then, on a similar theme, determine the amount of ripple or noise on those power lines to make sure the replacement has power supply rejection figures suitable (or maybe better than the current op-amp). Look at data sheet graphs for this. They should tell you PSRR figures across a wide range of frequencies. If the prospective op-amp doesn't have graphs don't use it.

Input common mode range and expectations of how large an output signal it has to produce might come next. Clearly, choosing an op-amp that has better input and output range is fine but, in reality this may limit choices so, an analysis of the circuit and studying the expectations of what might be asked of the op-amp are both important.

Gain bandwidth product is important to consider - does the prospective op-amp have enough to fulfil the gain and bandwidth expectations of the design? Has it got sufficient slew rate to deal with full range output signals at the highest frequency required? Again, you can just choose a better device but this (and all the other parameters) might limit your choices so, it's always a good idea to realize what the expectations of the target design are.

I'm not going to go into detail any further other than to make a list and all are generally important in one way or another: -


Input offset voltage (produces DC output voltage error that is "gain" x input offset)
Input offset voltage drift with temperature (can't be nulled out)
Input offset/bias currents and their drift with temperature (only of concern when resistors around inputs and feedback are medium to high value)
Low frequency noise figures and equivalent input noise densities for voltage and current
Open loop DC gain (basically produces a DC error - important when buffering voltage references that are meant to be extremely accurate for instance
Phase margin and Gain margin - produces ringing and possibly oscillation
Common mode rejection ratio
Gain stability in low gain configurations - see also phase/gain margins
Package options
Input capacitance (e.g. can produce filter errors and can create substantial HF noise)
Output current drive capabilities
Short circuit current (could be too high and destroy itself)
Output loading capabilities (including capacitance and overshoot)
Power supply current and power supply decoupling requirements
Settling time can be important now and then
Environmental temperature range
Differential input voltage limitations (if used as a comparator)
Dynamic output impedance in closed loop/open loop - poor op-amps in this respect might not make good active filters especially when not operated at unity gain.
Input signal inversion problems (quite a few op-amps will invert the input signal when inputs are too large in amplitude)
Maximum power dissipation
TotalHarmonicDistortion versus frequency (important for audio applications for instance)
Potential ambient light sensitivity issues (it can be a problem with many op-amps and in the IR region too)
Positive and negative overload recovery times


Hopefully if all the above are considered you should be OK.