Ideas

One of the main reasons young people don't participate in European politics is exactly because they are rarely given the opportunity to do so. One great idea would be to "reserve seats" in the EU parliament for young people- promoting the election and participation of people aged 16-25 years old. The EU must promote the creation and the election of youth political parties. Also, the creation and promotion of programs similar to Model United Nations, like a Model European Parliament, would get students excited to further research European policy and understand the sociopolitical factors that influence it.
Involvement in the democratic process and active citizenship must start at school. We cannot treat students as “citizens in waiting” throughout their entire education and expect them to know how to be active citizens once they turn 18 years old. This should start at school, if children are made part of the decision-making process, see change within their school and feel that their voices are heard on a daily basis, they will leave school knowing how to take action and believing that they can have an influence on society. Children spend around 12 years in school having all the decisions made for them behind closed doors –this could be a great opportunity to involve them in school issues (trips, timetables, holidays, the teaching programme, projects) through school wide voting, school meetings, committees, debates, projects that influence the school and are influenced by ALL students not just those in student council. I believe we cannot have a well-functioning democracy without democratic education in place. I would gladly write an essay about this 
One of the main reasons why young people do not have a voice in the democratic process is due to their low level of political participation in the form of voting and contacting politicians. For this reason, politicians are less likely to compete for the young people's vote in the elections. This is the circle that needs to stop. To this end, young people need to be encouraged to vote more actively through civil society groups where young people are present (youth forums, sports groups, school groups etc) and actively through the education system whereby a teacher explains the electoral importance and process. By doing so, a stronger voting turnout among the young will also attract politicians to engage with young people. In the context of EP elections, actual voting age should be reduced to 16 years old across the Union and the voting periods must be unionised whereby the MEP campaigns around Europe happen during the same time and where all Europeans vote during the same time.
People need to see their vote making a difference. Presidential elections, for example?
By improving and strenghtening the Erasmus project. I believe this measure would aid this good idea.
I also really like the idea of having randomly selected groups of pupils getting involved in local councils or something similar, a bit like jury service. (as someone suggested below)
It is a really hard question, and I know it is an issue for member states more than the EU, but we really need to advocate for basic, comprehensive political education which tells students about their rights in the EU. Knowing what the EU does for you, rather than taking it for granted as so many in the UK did, might help people feel more engaged - and the same goes for domestic politics.
One of the main reasons why young people do not have a voice in the democratic process is due to their low level of political participation in the form of voting and contacting politicians. For this reason, politicians are less likely to compete for the young people's vote in the elections. This is the circle that needs to stop. To this end, young people need to be encouraged to vote more actively through civil society groups where young people are present (youth forums, sports groups, school groups etc) and actively through the education system whereby a teacher explains the electoral importance and process. By doing so, a stronger voting turnout among the young will also attract politicians to engage with young people. In the context of EP elections, actual voting age should be reduced to 16 years old across the Union and the voting periods must be unionised whereby the MEP campaigns around Europe happen during the same time and where all Europeans vote during the same time.
What about local, regional and/or national Youth Commissions to serve as obligatory consultative bodies for decision-makers? I am not speaking about a twisted system where those who are liked by politicians get to participate, or youngsters aspiring to careers in politics. I am speaking about a system a la ancient Athens. We would have youngsters randomly selected to serve in the Youth Commission for a specific time, regardless of their social status or education and youngsters would receive compensation for that and any necessary support in terms of housing and transport if they have to travel.
By letting us participate in the first place! When I can't do anything because of lack of "experience", it doesn't help me at all!