Many practices are called meditation, but two types have been repeatedly validated by research: self-transcending meditation, and minfulness meditation. They both originated in the context of eastern religions, but are mental techniques without any religious component. Nonetheless, they are likely to enhance the experience of your own religion or philosophy.
Readily available versions of self-transcending meditation are TM (Transcendental Meditation) and NSR (Natural Stress Reduction). These use effortless repetition of a mantra (mental syllables without meaning) as vehicles for experiencing mental-physiological states of restful alertness and pure consciousness (traditionally called dhyana and samadhi).
Readily available versions of mindfulness meditation are MBSR (Mindfullness Based Stress Refuction) and Vipassana Meditation. These use effortless attention to breathing or scanning one’s body sensations as vehicles to states of broadened awareness and detachment from the ever-changing conditions of life (referred to in different schools as dhyana /jhana and satori).
As an observant Jew, I have practiced both forms of meditation for many years. I am a doctor of psychology, and a certified instructor of meditation and yoga with over 40 years teaching these techniques in TM centers, schools, and clinical settings. In my experience, the most important aspect of both types of meditation is that they teach one to gently attend to a simple mental vehicle (e.g. mantra, breath, or rhythmic motions) while allowing other thoughts and perceptions to come and go.
None of these methods aim to make the mind go blank, nor treat thoughts, noises, and sensations as distractions. They are accepted as a natural, even beneficial part of the meditation. When you notice them, you accept them as passing thoughts, and gently shift attention back to the mental vehicle (mantra, breath, etc). This shifting of attention between the vehicle and other thoughts can easily happen many times in a single 20-minute session of meditation. Additionally, one doesn’t try to control how the vehicle (mantra, breath, etc) is repeated — speed, clarity of pronunciation, etc. will all change. One allows these features to shift effortlessly. Indeed, at some points in meditation they may completely fade out, disappear, or stop for brief periods. All of these changes are signs of correct, effortless practice.
Research indicates that both types of meditation lead to novel states of mental flow and altered physiological functions. The precise physiological states differ somewhat between different techniques, but they all tend to promote states of reduced stress and increased awareness, via minimal effort.
Regarding your question about mantras: I have extensively studied traditional practices, eastern and western, and I can assure you that the mantras used by TM and NSR are not the names of any gods, teachers, holy places, or rituals (et cetera) that I have ever seen or heard. Just as these meditation programs state, their mantras are pleasant sounding syllables that do not have any recognizable meaning. I would add that these meditation methods do not really emphasized the mantra as the main point; they are just tools of meditation.
If, despite these assurances, you are still cautious about mantras but still want to try the method, that is okay. Instead, you may want to choose a short, pleasant word from your own tradition. The most important thing is that you find the sound harmonious, and that you learn to use it in the effortless manner described above. Some people find this easy to learn on their own. However, you may have greater confidence if you find an experienced meditation teachet to guide you to the essential effortlessness of practice.