What is the ruling on a girl aged 18 naming herself on Facebook "Strawberry Biscuit," and is there any objection to it?

The default rule is the permissibility of using pseudonyms. However, a woman should avoid using them in contexts where non-mahram (definition) men may see them, so as not to tempt those with weak souls. Similarly, a Muslim woman should avoid entering mixed websites and forums, and limit herself to beneficial women-only forums; this is further removed from suspicion and safer from temptation.