Are the verbs that are conjugated to end in "-n" in the past participle related?

This is a simple question, which actually requires a quite complex answer (which I've made as simple as I can). This is because there are several phenomena at work here.

1. English is of West Germanic descent, as you well know, and beyond this descends from Proto-Indo-European.

2. Old Germanic has 2 classes of verbs, strong verbs and weak verbs, which are differentiated by the way they form their past tenses.
   Don't ask me why they are named like this; those weak/strong names come from a German guy called Jacob Grimm. Being German, he called the phenomenon of changing stem vowels in inflections "ablaut". More broadly, vowel changes like this are common in Indo-European languages and not restricted to verbs (man => men; Man => Männer; homō => homin-ēs).
   Here are a few examples for German verbs (using modern forms for the sake of clarity).

   • The weak verbs are the ones that keep their stem unchanged and add a dental ( a 'd' or a 't') at the end. For instance:
     lieben (to love) => Ich liebte (I loved) => geliebt (loved).
     Old Germanic and Old English had four classes of weak verbs, depending on morphological rules.

   • The strong verbs don't do that at all. Instead they change the last vowel in their root.
     For instance:
     singen (to sing) => Ich sang (I sang) => gesungen (sung)
     Old English and Old Germanic had seven classes of strong verbs depending on the rules used to build the past tenses.
     But look how similar the German and English vowels actually are.
     This is in fact a general rule: if you know your German strong verbs, learning the English irregular verbs is much easier (or the other way round if you're learning German).
     So these are the verbs that are more likely to end with an "-n" in English.

3. The origin of the weak dental suffix is the Old Germanic suffix "ōdaz".
   In Old English it gradually transformed in "-ode" (or "-ade" or "ede" depending on the dialects"). In Old Germanic, this "oda" can be considered as an ancestor to "did" so that to say "I loved", you would actually say "I love did".

4. After the migration to Britain, German and English followed separate evolution paths. and the "odaz" was transformed in the German "-te" on the eastern side of the English channel and "-ed" on the western side (for the preterite).

5. Another phenomenon has to be taken into account. Irregular verbs tend to disappear. The less commonly used the quicker their "regularisation". This is because of "incomplete grammatical education" when old morphological rules are supposed to be passed on to newer generations.

6. All recent verbs are regular and there are in English a number of irregular verbs that are currently in that process of becoming regular.
   You can observe that they admit both a regular and an irregular past tense: strived instead of strove is a well known example.

So, the difference you note started in Old English, which had the concept of strong and weak verbs. Strong verbs marked their past and past participles by varying the vowel sound in the base of the word (e.g. swim/swam/swum in Modern English). Weak verbs used a suffix (-d or -t if I recall correctly, feel->felt is a modern example). Through Middle English, new verbs that were added used the weak verb paradigm and this approach became the foundation for what we know as 'regular verbs' now.

One class of the strong verbs maintained the vowel shift only (sing/sung, drink/drunk, ring/rung). A second class of strong verbs uses a -(e)n suffix. This is the class you're referring to: arise/arisen, eat/eaten, throw/thrown

Edit: I'll also note that in the second class, the vowel shift in the past form is often maintained, e.g. throw/threw/thrown, fall/fell/fallen, grow/grew/grown.

Here is an exhaustive cataloging of verb forms you may find of use: David Appleyard's Quest for Regular Irregularity in English Verbs.

As Appleyard says in his intro:

English belongs to the Germanic group of languages descended from Proto-Indo-European. One of the characteristic features of Old Germanic was its utilization of root vowel inflections to indicate changes in verb tense. While modern English makes use of a simpler suffixing system (e.g. call-ed, phone-d, etc.), verbs with irregular vowel inflections still survive for many of the oldest and most basic activities known to man, such as eating, drinking, sleeping, sitting, standing or swimming. This means that even the most elementary of students soon have to get down to the nitty-gritty of committing them to memory and I would very much like to facilitate this process.