It depends if their mother is the same person. If Lily and Jenny are sisters and have the same mother then it is 'their mother has...' If they have different mothers then it is 'their mothers have...' A quick correction. 'I have friends. Their names are Lilly and Jenny.'

First of all it should be "I have friends. Their names are Lilly and Jenny". You need to use the phrase "their names" because there are more than one! You can say "her name is Jenny, and her name is Lilly" or you can say "their names are Lilly and Jenny". Both of your examples are correct, they just mean different things. Please look at this: 1) I have friends. Their names are Lilly and Jenny. Their mother has brown hair. 1) I have friends. Their names are Lilly and Jenny. Their mothers have brown hair. Sentence number one means that Lilly and Jenny have the same mother, so they must be sisters. Their is only one mother, so the singular is used. In sentence number two, there are two mothers. One is Lilly's mother and the other is Jenny's mother. In this case, since there are two mothers, the plural is used. Hope this helps!

Their mothers have