In many situations, there is little benefit in a race to the courthouse. Sometimes, however, it can make quite a difference who becomes the Personal Representative (executor) in a probate.
There may be several people who have an equal right to become the personal representative (i.e. the executor). For example, if there is no will, and the decedent has no spouse, but has two children, whichever child is the first to file a Petition for Probate is likely to become the personal representative.
The personal representative is a fiduciary, and must do what is best for the estate (including the creditors, and the beneficiaries). However, there are many times when being the personal representative can confer a lot of power that can be very difficult to challenge. Examples can include whether to sell an asset, and to whom an asset should be sold, how various items should be categorized (which can change who gets the asset), and even whether or not to bring a law suit alleging elder abuse against a former caregiver (who might even be one of the persons with the right to be the personal representative).
In addition, in some circumstances, the first person to file for probate controls which county will handle the probate. This can make quite a difference if there is the possibility of doing a probate in La Grande, where the decedent held real estate, or in Corvallis, where the decedent was when he died, for example. This is especially true if there is a conflict between two potential personal representatives, and one potential personal representative (and his or her lawyer) lives in La Grande, and the other is in Corvallis.
A personal representative can still be removed for cause, if the personal representative commits various bad acts (including telling the court he has priority to serve as personal representative, when actually someone else had priority). However, without some good cause to remove the personal representative, the first person who files, and who is legally entitled to serve as (or nominate) the personal representative will likely be appointed, and will likely then control the probate.
Similarly, the court where the first person who is authorized to file does file is likely to be the court that will hear and control the case. This is true even if some other equally entitled person files in some other court before any notices are sent, or the like.
Some years ago there could be a question when a case was actually filed, since many courts only date stamped a filing. When there was a race to the courthouse, in those days, the lawyer would get the court clerk to add an actual time as well. Now the time of filing, to the second, is automatically added for every case filed in court in Oregon, so these disputes are easily resolved.
If there have been two cases filed, the side which filed the second case generally is responsible for dismissing the second case, since there is already a case pending.