Should I use 'was' or 'were' in this example?
I was always delighted when my brother or one of my sisters was/were asked to do them.
This is one of those situations where English syntax has not yet come up with a good rule. Usage varies, and nothing really feels good; so, while conjoined subjects with varying number are additive
disjoined subjects (with or instead of and) with varying number are usually avoided, for precisely this reason.
There are other situations, as well; note
A good reference on the subject is Jerry Morgan's 1972 paper "Verb agreement as a rule of English", in Papers from the Eighth Regional Meeting, Chicago Linguistic Society, pp 278-286.
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The rule in English for when you have disjunction in a compound subject (meaning two of more separate subjects connected by an “or”, or by a “either ... or”, or by an “neither ... nor”), is that the verb agrees with the nearer subject — or nearest, if there are more than two.
The compound disjunctive subject here is “(my) brother” on one side and “one (of my sisters)” on the other. The nearer of the two subjects is “one”, which is as archetypically singular as singulars can ever get: one is always one; one is never many.
Therefore, the number of the subject is singular, and so too must be the number of the corresponding verb. That means that (since we’re not in off in the hazy land of Subjunctivania here :) the verb form needed here must be “was”:
Edited to Add
Compound subjects of several things all added together are always treated as plural. And when you have a compound subject of disjunctive (or- or nor-separated) things that each have the same concordance, such as in this example, it’s still not so bad.
However, when you have several elements that each would take a different verb form, the formal rule in English about choosing whichever subject sits closest to the verb for that verb’s concordance can result in something strange that many writers, and surely most speakers, would rewrite to avoid the awkward situation.
This is perhaps most readily demonstrated using the verb be, because of its wider variety of distinct forms in both present and past tenses.
In older times, these were even more apparent, since we had thou forms. Percey Bysshe Shelly has a bit of verse where he uses “that or thou art”, so you can see that choosing the nearer subject for concordance isn’t some new notion in English:
Nonetheless, many of those can make some people uncomfortable. In casual speech, some people simply switch to the plural no matter what formal logic may demand. If you are someone who construes neither to be plural, this may work for you. If you are not, it probably will not. And even if you get away with it in speech, in a written medium, you may wish to avoid such a thing. You’ll have to decide for yourself which of these pairs sounds better/more correct/less bothersome to you:
I can’t bring myself to say “neither are” or “one are”, but some people can. However, if you are one who does — or perhaps in this case, one who do :) — then you may be able to live with an unfailingly plural concordance even of disjunctive singulars. For example,
I wouldn’t say any of those myself if I could possibly help it.
In casual speech without the luxury of time, you may hear things said that could be frowned upon in more formal circumstances.
In writing, you can (and arguably should) take the time to rearrange things so that you don’t have to choose something that fails formal concordance requirements. It’s not a bad idea to rewrite it so you can avoid sounding awkward no matter how “correct” it may be. Awkward and right is still awkward.
I would use was, as the subject of the clause—
— would be singular: either there is one brother who is asked, or one sister who is asked, but the phrasing explicitly excludes the possibility of more than one sister being asked. Absent any context, I might further infer that only one person can be asked, negating the possibility of both the brother and one or more sisters being asked, and again keeping the verb singular for agreement.
Things could be more ambiguous if phrased differently, as noted in John Lawler's answer.
Let us consider 1. I was always delighted when my brother WAS asked to do them : obviously correct 2. I was always delighted when one of my sisters WAS asked to do them : obviously correct 3. I was always delighted when he or she WAS asked to do them : obviously correct
It follows that WAS is to be used in this case.