First of all, Lighthouses are an attraction. Most of them are not any longer in use. They can be visited from outside only, occasionally from inside. Some of the Lighthouses have a dramatic location, some a touching history and others are simply beautiful.
The tourist season on Lampedusa starts in the beginning of June and ends in September. During this time, the island is a bit crowded. From my experience, it is best to visit the lighthouse early in the morning or in the evening. During this time, you can have the place just for yourself.
Texel where the Eierland Lighthouse is located is a year round destination. In summer the lighthouse is open to the public daily. Autumn and winter are less crowded.
To beat the crowds visit between arrivals of the ferries. During winter only open on Wednesdays.
Best time is sunset, less crowded, absolutely stunning and picturesque. Avoid weekends, it is also very popular by the local people. Very special in May and June when the hills are covered with flower carpets.
Every Friday in July and August you can climb up the stairs between 7-9 p.m. You get the tickets at the boathouse Boulevard Zuid 5. I like it most with dunes infront, so late afternoon until evening is the best light for this picturesque Lighthouse.
Peggy's Point is world famous and very popular. Tour buses and rental cars arrive throughout the day especially in the summer. However, there are two easy ways to beat the crowds even in the peak months:
Be there in the morning as early as possible. At least before 9 AM. Experiencing the sunrise is amazing anyway.
Don't leave before or right after sunset. Just stay! Most visitor leave when the sun goes down, but the place gets even prettier. Check sunrise und sunset times here: Halifax Canada | Sunrise - Sunset