Things to do in Brighton, for kids
Most of Brighton's obvious children's attractions lie along the seafront - notably the Palace Pier and nearby seafront play areas. There's a "crazy golf" course by the pier, at the foot of the Brighton Wheel, and on the other side of the wheel is the Aquarium Station for the Volks Electric Railway, which offers an engaging ride along the seafront, while sitting down (so that the little ones can't complain that they're tired). Riding the Volks feels a little like riding a fairground "ghost train", and it's a nice way to see the promenade architecture, but it only runs in summer months (and is sometimes halted when things are too wet, or too windy).
At the end of the Volks line is Black Rock Station, which doesn't offer a great deal to do other than to look at the sea and maybe have some lunch before heading back, but halfway along the line is the Peter Pan playground area.
Also by the pier is the Brighton Aquarium (which is like an underground dungeon with fish), but it's a little pricey unless you book ahead.
Further from the pier
Inland from the pier is the Pavilion, and the pretty Pavilion Gardens, which make a nice place to sit down for a break. Also in the Pavilion Gardens is the Brighton Museum and Art Gallery, which is free, but perhaps not so exciting for children (under-tens often aren't interested in displays of art nouveau furniture, no matter how good they might be).
For slightly older kids, there's a Fishing Museum a few minutes west of the Palace Pier, halfway to the burned-out skeleton of the old West Pier, which isn't just about fishing - for instance, they also have a display about Punch and Judy.
Back from the seafront (under Brighton Railway Station) is the Brighton Toy and Model Museum, and westward into Hove is the Hove Museum, which has a lot of child-friendly exhibits, but is a fair way away from central Brighton and perhaps not ideal for daytrippers because of the location.
Another interesting "educational" location is the Booth Museum (aka "the dead bird museum"), which has huge numbers of birdlife tableaus from bluemartins to eagles, and also has a surprisingly thorough collection of animal skeletons, and a "rocks and minerals" display. Unfortunately, the Booth is probably more than half an hour's walk from the pier, so it probably needs a special trip.
Bus services in Brighton are good, so with some advance planning, it's not too difficult to get to some of the more outlying attractions. Car parking in Brighton tends to be problematic.
Seasonal attractions and events
Brighton hosts a huge number of festivals throughout the year, many of which are aimed at children or have child-friendly elements, such as the Brighton Science Festival and the Childrens Festival. There are also a number of nice parks (such as St Anne's Wells Gardens) that organise events, but you'll probably need to check the dedicated "listings" websites to find what's currently on.