I see a lot of good advice here. Highschool isn't going to have overwhelming turnout. But I think you can get more people involved.
1. Make the club useful for other people. Have workshops on wordpress, blogging, have career speakers maybe from local colleges. Have discussions on 3D printing to bring in the DIY crowd. Computer Science means you have access to all the information and systems that have ever existed. It's core is founded on helping people do things they want to do and make their lives easier. Think about it that way and you'll see that you have plenty of material generate interest.
2. Bring pizza! People often skip meals to make meeting times. Understand that and use a large portion of fundraiser money for pizza. It works.
3. Get lists of emails/text and send them weekly updates with really useful information.
You have "old" computers available. Buy 15 licenses of quake 2 for a few dollars and hold a game-off! Have fun! But don't make the club about the computers. Clubs are about people and what your interest can do for people.
Myself, I was a former CPSC Club president and a UPE president, I can tell you what worked and didn't work in college for me.
In college successful meetings involved:
Going deeper then just one layer of meetings and make it a club. Making it something useful.
Advertise! Chalk sidewalks, make chalkboard notes in cpsc classrooms, put up flyers, email lists, reminder emails, phone calls sometimes. Have a big pizza meeting and get a group list to text. Keep people thinking of your club with signups for updates on related campus events and important industry changes, like updates to google posted on searchengineland. Updates on the most popular programming languages/web systems with reports from google trends. Talking about trends students need to know.
Holding meetings at Lunch times or common hours for the best turnout.
A big key for meetings is food. Free Pizza / soda / fresh coffee can be paid for by coffee pot fund raisers, sometimes alumni and other general fund raisers. The students must have food, even if it's just a 75cent slice of pizza and a 10cent free cup of coffee. This makes up for the lost lunch/break time students use to go to your meeting.
Hold the meeting in an accessible public location like a central hall with lots of chairs.
Speakers from industry often go over well and so do discussions of student projects.
DIY robot demonstrations,
Speakers on server/web security,
Discussions on getting experience though open source projects,
Book reviews on the best book for X,
Best twitter/feed resources, project management softwares, specific topics like content development...
More modern topics like social media, VR web apps, phone app programming, 3D printing, computerized stencil cutters, factory robots and google glass stuff would all go over well too.
Make your meeting important and worth going to! What were the 10 most important articles/web trends on slashdot in the last year?
If the speaker has a printable guide say on networking, have printouts or sharable files you can email to people who sign up for a copy. Your adds for the events should say there will be guides available at the events.
Talk with professors to help arrange speakers, they often know a few people. And try to get them asking these people about internships. Then you can help arrange internships for students. Also saying on a flyer you'll be announcing internship opportunities will also bring a lot of people in.
Speak but don't spend too much time talking yourself unless you're presenting or if you can magnetize an audience. Ideally spend your time helping speakers talk and come with a few guiding questions for your speaker.
Have one page handouts with a summary of the topic and the next meeting/time/schedule. Have a texting/email list signup for people and mention that to them.
Post meeting minutes and event summaries online, with links to documents/resources if available.
Announce meetings and club events that are related, like web graphics design discussions or writing for the web workshops or resume workshops. Have a writers workshop to introduce them to web resources and writing resources like tvtropes! Hold Saturday morning workshops as part of your club and to make your club useful. Help organize study groups, computer help, programming contests, note sharing, career planning and problem solving workshops. Hold a web publishing workshop and use it for inviting people in different majors to a social. Hold a club meeting looking for volunteers to holding workshops (for resume experience) and for getting lists of people for inviting to a social. Have a robotics workshop and coordinate with the art department as they often have tool shops and DIY enthusiasts. Don't just hang out! Make connections!
Have field trip events, like paid/subsidized field trips to science museums. Everyone loves a big group trip at the end of the semester.
Organize multi-player computer gaming events with pizza. If you have everyone together and have breaks from the games, it becomes a social.
Once a month coffee & coco & cider fundraisers in building lobbies worked pretty well. But don't hold them too often because most people are buying coffee/snacks to donate to your club and won't buy more often then that.
At the meetings, recruit people who would like to help with fundraisers. Mention that fundraisers and club involvement are good resume experience. That tends to wake a few people up.
What didn't work:
No Potato chips, popcorn, cookies or other cheap junk food. Also bad locations, like a lower level auditorium won't give you a good turnout. Computer labs aren't the best place for meetings either, they are distracting and are in use when you need the space. Classrooms aren't good either because students are used to them being used by a teacher.
Multi-club socials are a challenge. Like getting the Math club involved won't work. We take a lot of the same courses, but at my school "pure math" math club students were too young and stuck up to hang out with "applied math" computer science losers. Honors clubs (besides maybe UPE) are also often stuck up. At my school the main honors group was trying to make that group women's only. So they had lots of women, but they had a had a political issue with groups that have a lot of men and I couldn't arrange a social with them either. So it's good to try to involve everyone, but for a good social don't spend too much time trying to involve groups with a chip on their shoulder.
How to make socials work:
Talk to some art or writing department heads. Art/Graphic design clubs are less stuck up and more flexible minded, and are often involved in web design as a commonality and they have more women, so if you want to have a multi-club social, that can work. Also writers are sometimes a good mix, as writing and web publishing have become more closely tied. A "web publishers social" or a "design online social" can work. A "web marketing social" could also work. Just think about commonalities you have with groups that don't a chip on their shoulder.
In the club, deal with problems. If a secretary isn't showing up, not posting notes online, not putting up flyers, depressed, uncle died... Address the behavior as a group and ask if rearranging things makes sense. The person may not go and they may create drama, but try not to make it dramatic yourself. There will be people elected who aren't right for the position or are too busy spend the time. Just rearrange things and find volunteers to do the work that's being slacked on.
Also sometimes speakers don't show up. Come with a plan B speech on your favorite topic X and roll with the punches.
Ok so a lot of this might not apply to your high school Computer Science Club. But I hope this gives you some ideas and helps you realize just how helpful a computer science club can be for your school!