Seems to be very rarely used, as pieanne pointed out. However, here is one instance: Assuming the tribunal would not be so careless as to make such an oversight, it is submitted that ... (Source: Journal of the Law Society of Scotland. Edinburgh: Law Society's Hall, 1975, pp. 115-150.)
Use "do" for general activities like "the ironing", "the washing up", "a job". Use "make" for constructing, building, or the opposite; thus "make a cake" "make a cup of tea", "make a plan", but also "make a mistake", "make an excuse", "make an exception". The word "oversight" is like a mistake. Thus, if you must use it in the active voice, "make an oversight" would be correct. But as other commentators have indicated, a turn of phrase such as "there was an oversight" would be more usual.
An oversight is not so much an action as a failure to perceive: so 'do' is not appropriate.
'Make' is the more general term.
Hoever, if 'oversight' must be the direct object, then the best verb is 'commit': 'He committed an unforgivable oversight.'
Or'He was guilty of an oversight', 'It was an oversight', etc.