It usually doesn't take too much time to convince a camper to move into the town, provided that you have enough room.
After the camper asks you about the town and perhaps whether or not it'd be nice to live there, any game that he or she asks you to play has the chance to be a game that determines whether or not that camper moves in.
For example, I recently found Maple camping in town, and the very first game she asked to play was one that involved guessing directions, saying that she would move in as part of winning the game.
To speed up the process, you can simply refuse any game the camper offers that isn't related to moving in, and repeat if you happen to fail one of these games. Since the camper hasn't moved in yet, none of these actions will offend the camper, and you are free to do this as often as you need to until the camper is convinced.
There's no "punishment" for letting villagers go, the main gameplay reasons to keep the village "fresh", new people, new experiences, new furniture (the items a villager gives you is often related/from their interior decorating).
There is however the issue of character portraits. If you are nice to a villager for a long time they will eventually give you a portrait of themself which can be used as furniture. Each is unique to each villager and counts as a lucky item, good for large amounts of HHA points and (I believe) some feng shui.
The only way to get all villagers' portraits (an extremely daunting task likely requiring multiple years of daily play) or, more realistically, to get portraits for the villagers you like, is to keep villagers around until you get their portraits and send enough villagers packing to rotate in new villagers who's portraits (or company) you want.
Also note that the personalities of villagers (snooty, sport, smug) determine which public works projects they can request, so to unlock certain public works projects you may HAVE to let some personalities leave so that others of the required type move in.
Generally speaking, let villagers go if you only have a passing attachment to them (or none) and only keep a couple villagers "forever". Keeping too many villagers forever will keep gameplay a bit stale; you won't see all the personalities, species, individual characters or the interiors of their homes. Since it's somewhat rare (one a week in my experience) for them to ask if they should leave, I would recommend taking them up on it unless you really want them to stay, to allow for maximum variety.
Note that many villagers moving in/out will possibly replace trees or flowers (I don't believe their homes ever remove public works projects or rocks). If you have a Police Station this is less of a deal as the items removed will become "lost" in the lost and found at the police station.
Best Answer
Yes, you did convince her to move in. What I think she meant when she said that you have to convince your other villagers that she would be cool in your town was so that she would have a good reputation when she moved in.