How much would it cost to replace an alternator on a 97 Honda Civic LX?

Labor should be just under an hour in the Mitchell labor guide (around .9) with a standard dealer rate of $75.00-$85.00 per hour.

The alternator varies in price, usual list price on the alternator is $279.68, you can find this cheaper, or have it rebuilt much cheaper if the bearings are in good shape as usually the regulator, stator, and brushes come in a kit.

Doing this yourself is quite simple and requires 10mm, 12mm, and 14mm sockets and wrenches, it's completely accessible, however remember always disconnect the battery before performing any operation that involves electrical wiring of any sort.