I assume you have the gene cloned in an expression vector and have a functional assay established to access its activity. You could use site-directed mutagenesis but it may be too much work and too many sites to test. And worst of all, you may end up not finding the most critical or all of the critical residues of the protein, if any. Random mutagenesis would allow you to create a mutant library blindly and screen for the mutants that show varied activity. Subsequently you would sequence those that show interesting activities, be it loss or gain of function, to see which residues might be critical to the function of the protein.
For random mutagenesis, you could use error-prone PCR-based method to create the mutant library, which involves the use of mutagenic PCR to amplify the gene, cut and paste back into the vector for expression. However, if the gene is large this method is very inefficient and the mutant library is often not fully representative.
We have a new random mutagenesis reagent kit and method, AquaMutant, which is chemical-based. You may either soak your cut out gene with AquaMutant to introduce random mutations or you may treat the amplified PCR product with AquaMutant to introduce random mutations. The mutation rate can be varied and controlled by the concentration of AquaMutant (from 1-100%) and the exposure time (from 1-30 min). Subsequently you would clone the mutated gene back into the expression vector.
If you hate the cutting/pasting cloning workload, which can be time-consuming and frustrating, you could try simply soak the isolated plasmid DNA directly in AquaMutant to get the mutant library. Of course, you would have mutants with mutations in the plasmid other then the gene of interest and you would lose mutants with mutations in the antibiotic resistance gene or origin of replication, but you can still get a library that is more than large enough and fully represented due to the use of micrograms of plasmid DNA for the random mutagenesis.
In addition to the above in vitro random mutagenesis approaches, you could use AquaMutant to treat the cells harboring your episomal expression vector in an in vivo random mutagenesis. You would then use FACS sorting to pick out and separate those cells expressing your protein for activity assays. You don't need to care about those junk mutations elsewhere in the cells for the moment. After you have sorted out the interesting clones, you could re-introduce the episomal mutants into the original cell line to confirm their activities. The options are all yours.
Best of luck!