Ali had two sons, Al-Hasan and Al-Hussain, so why do Shia care so much about the Al-Hussain rather than Al-Hasan?
I often hear them chanting
Labayeka Ya Hussain
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There are several reasons for this, and definitely not that Shia value Al Hussein more than Al Hassan.
First please read here about hadith Al kisa. In short, the hadith is about the gathering of 5 members: The Prophet (PBUH), his daughter Fatimah (a.s.), Al Imam Ali (a.s), Al Hassan (a.s) and Al Hussein (a.s).
This Hadith shows the importance of these five:
Then the Lord, Almighty Allah said:"O My angels! O Residents of My Heavens, verily, I have not created the erected Sky, the stretched earth, the illuminated moon, the bright sun, the rotating planets, the flowing seas and the sailing ships, but for the love of these Five lying underneath the cloak"
We know that the prophet was the first one to die from these five, then Fatima then Al Imam Ali (a.s.) then Al Imam Al Hassan then Al Hussein.
When the first four of these died there was at least one there that reminds people by the importance of these five. But when Al Hussein died, he was the last and no one else was there to remind people by these five. So his death resembled the death of these five together and hence its importance.
Here Is a hadith from Imam Al Sadik (Jafar al-Sadiq) that supports the above statement?
Another reason I will add here is that his death resembles the Revolution of right over wrong. The mourning of Al Hussein holds deeper meanings than just crying about his death it is a way to remember to stand with the right against wrong and to remember the sacrifices of the prophet and his family.
To clarify, it doesn't make any sense that Al Hussein was better than the prophet or even his brother Al Hassan. It is just that the death of Al Hussein holds deeper meanings than just his death it holds many meanings and one of them is remembering the sacrifices of the prophet as he said:
Hussein is from me and I'm from Hussein.
As per request in the comments here is the isnad of hadith al kisaa:
،قال البحراني :"رأيت بخطّ الشيخ الجليل هاشم البحراني عن شيخه الجليل السيّد ماجد البحراني عن الشيخ الحسن بن زين الدين الشهيد الثاني العاملي عن شيخه المقدّس الأردبيلي عن شيخه عليّ بن عبد العالي الكركي العاملي عن الشيخ عليّ بن هلال الجزائري عن الشيخ أحمد بن فهد الحليّ عن الشيخ عليّ بن الخازن الحائري عن الشيخ ضياء الدين عليّ أبن الشهيد الأول العاملي عن ابيه عن فخر المحققين عن شيخه المحقق عن شيخه ابن نما الحليّ عن شيخه محمد بن إدريس الحليّ عن ابن حمزة الطوسي صاحب "ثاقب المناقب" عن الشيخ الجليل محمد بن شهر آشوب عن الطبرسيّ صاحب الإحتجاج عن شيخه الجليل الحسن بن محمد بن الحسن الطوسي عن أبيه شيخ الطائفة الحقةعن شيخه المفيد عن شيخه ابن قولويه القمي عن شيخه الكليني عن عليّ بن إبراهيم عن أبيه إبراهيم بن هاشم عن أحمد بن محمد بن أبي نصر البزنطي عن قاسم بن يحيى الجلاء الكوفي عن أبي بصير عن أبان بن تغلب عن جابر بن عبد الله الأنصاري رحمهم الله تعالى جميعاً أنَّه قال
Your inference is totally wrong. We Shia care(respect) equally for all the members of the household of prophet
"Labbaik-e-ya Hussain" is basically an expression in times of battle while facing the enemy or facing any kind of injustice from tyrants. Being 'at the service of Hussain' expresses the idea of fighting for justice and being on the side of justice.
It's a slogan of believers who worship only Allah, The Almighty, and thus they support the righteous slaves of Allah. And Hussain was one of the righteous slaves of Allah.