Many English speakers believe that using the plural pronouns they, them, themselves, and their in gender-neutral singular constructions is incorrect. For example, these people would consider the them in “call a friend and ask them to come over” to be wrong because them by definition refers to multiple people, whereas in this clause its antecedent (a friend) is singular.
But there are problems with this view. The main one is that English needs singular gender-neutral pronouns. There is no way to completely avoid them, as they are required whenever we need to speak about an unspecified person whose gender is unknown, and this sort of situation is far from rare. It’s always possible to reword sentences to avoid the problem completely, but this tends to result in unnatural-sounding constructions, and it requires you to waste a disproportionate amount of effort solving a problem that doesn’t actually exist.
There are three main fixes for English’s lack of singular-gender neutral pronouns. The first is to use either he or she and trust the reader not to be confused. This was the approach for centuries, though of course writers traditionally used the male pronouns, a practice that rightly came to be viewed as sexist by the 20th century. One can avoid the sexism by using only female pronouns or by alternating female and male, but while this makes the option slightly better, it doesn’t change the fact using the male or female pronouns this way can cause confusion. Plus, the practice tends to call attention to itself. The reader stops and thinks, “Oh, this writer is doing the she–as-gender-neutral-pronoun thing.”
The second option is the one that now prevails in much formal writing. It is to use he or she, him or her, himself or herself, and his or her. This approach is conventional now, so it is the safest choice in writing for school or work, but it is not a permanent solution. The clunkiness of the phrases goes against the modern trajectory of the language, which is toward breeziness and concision, and the practice calls attention to itself similarly to the first option. It also tends to sound formal, causing many people to shy away from it in speech and in less formal writing.
The last option is to move beyond the old view that their, them, themselves, and they cannot function as singular pronouns and to embrace them in this use with no qualms. This is tricky because anyone who takes this approach must accept that a certain percentage of their readers are going to think it’s simply wrong. But besides that, there is much to recommend this approach. Most important, the singular their, them, etc. are already widely used, and they have been widely used for centuries. Many English speakers use them unconsciously without ever questioning the practice until someone calls attention to it. In fact, the supposed problem of English’s lack of singular gender-neutral pronouns has already been solved. It’s just that some people have been led to believe that a perfectly natural habit among a preponderance of English speakers is wrong.
So don’t hold back. If you think the singular their, them, themselves, they sound fine, don’t stop yourself from using them just because some people think there is a rule.