In general the answer to this question and all similar ones is to first identify the chipset and current processor. For that CPU-Z is often helpful.
Now we have some basic information what to look for.
Next we should note the important characteristics of the CPU. They are type of socket used, code-name and maximum thermal power dissipation. Right now, the most important is the socket. Use for example wikipedia to get information about the socket. Usually we can't replace CPUs in ball grid array, so this is where the process would end in our case.
This combined with chipset will give us a list of CPUs that would be possible candidates as a replacement.
First we sort the CPUs by socket and remove from list any that use other socket types than our CPU.
Next we sort them by max TPD and remove from list any CPUs that hae greater TPD than our own. This is to make sure that the laptop will be able to keep the CPU cool. THe CPUs with lower TPD are of course welcome.
After that comes the tricky part. We now have a list of CPUs that could reasonably be used in the system and we need to make sure that they are actually usable. Some may require newest BIOS version and some may not work at all.
So now we need to look for different configurations of the same laptop and see what CPUs are available in them. This may or may not be difficult and mainly depends on searching capabilities. There are good chances that any CPU on that list will work fine. If none of the CPUs which were used in different configurations are interesting, we should search the Internet to see if anyone had any experience using a particular CPU we want to use and the laptop. This information can be used as an indication if the CPU will work well or not. If no information is available, we should take a look at the code-name and release date. Often what is basically same CPU will be released in several different versions. We can expect that a more powerful version of the CPU we're using will work fine (but in that case it would also probably turn up in previous searches).
If we can't find any interesting CPUs even after this, then the last resort is to experiment and try a CPU of the same generation and hope it works. There's a chance that it won't so it would be wise to have some backup plan what to do with the CPU if it doesn't work.