My apple cider vinegar became infested with fruit flies and maggots. Can I wash the mother and use her again?
Yes, as far as personal use is concerned, but it would not be recommended by food safety professionals.
I'd suggest instead cutting out a small piece of the mother and making sure that it is completely clear of eggs, or you'll repeat the experience even with washing.
Vinegar is known to inhibit bacterial growth across a wide range of foodbourne bacteria (more scientific version here), so the insects are unlikely to have corrupted it unsalvageably.
Further if the vinegar mother is bacterially contaminated, the 3-week period of fermenting the new vinegar will show it well before you use the vinegar.  Just be very wary of any off smells or colors.
Note that if you were asking this in a food service context you absolutely would need to throw everything away.  The risk of contamination is small and not worse than many other things in home kitchens.  However, the standards for professional cooking are higher.