Why is "Son" a short and simple word, but "Daughter" longer with a lot of silent letters? Also, does the -er ending have etymological significance?

Great question! Before we go any further I want to give a quick shoutout to the fine folks at /r/linguistics. It's an amazing subreddit which is generally better suited to these types of questions (duh) but I believe there are some linguists on the payroll in this illustrious place. In lieu of an actual linguist, however, you'll have to deal with yours' truly today... just some random guy with an interest in linguistics, a chip on his shoulder and more time than he should have at 5AM in the morning. Think of me as your hungover, slightly pervy substitute teacher, or your step dad who'll never be as good as your real dad. Make of that what you will. Now then, let's pluck this duck.

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I'm going to split your question into three separate parts. Firstly, why is the word "son" so unlike the word "daughter"? Secondly, why does the word "daughter" have so many silent letters? Finally, does the ending -ter have any etymological significance?

To answer the first of these we need to give a tiny little background in European historical linguistics. Essentially, without going into levels of depth unnecessary for the purposes of this response, the vast majority of languages in Europe, Greater Persia and the Indian subcontinent derive from a single "proto" language. A proto language is a language which can be supposed, usually though a series of very complex linguisticy shenanigans called "the comparative method", to be the most recent common ancestor of a family of languages. Just like how, for example, an as of yet unknown ape-like creature is the most recent common ancestor of humans and chimps, so too do languages have ancestors and descendants that spread across history in exactly the same way a family tree would do. Apart from creoles. Creoles are weird.

The proto language in question is Proto Indo European (hereon referred to as PIE). It is the most recent common ancestor of languages as diverse as English, French, Russian, Farsi and Hindi (among many, many others) and was spoken roughly four to six thousand years ago. In PIE, the words for "son" and "daughter" were very similar. Daughter has been reconstructed as *dʰugh₂tḗr (don't worry too much about the weird symbols, the study of PIE is an academic pursuit and so it requires weird symbols to make linguists feel special). Son has been reconstructed as *dʰeh₁ylios. Both of these originally meant something like "the suckler", stemming from a root *dʰeh₁ meaning "to suck".

So, with that in mind, where did "son" come from? *Dʰugh₂tḗr looks very much like the word "daughter" but "son" and *dʰeh₁ylios are clearly nothing alike. Well from PIE spawned a bunch of other languages, some of which went on to become proto languages in their own right. Some of these are Proto-Germanic (the most recent common ancestor of the Germanic languages; English, German, Swedish, etc), Proto-Celtic (the ancestor of Irish, Gaelic, Welsh, Breton, etc) and Proto-Indo-Iranian (the ancestor of the many Iranian and northern Indian languages, Farsi and Hindi being the most widely spoken of these respectively). In Proto-Germanic, the PIE word *dʰeh₁ylios (son) was replaced by *sunuz, itself from PIE *suh₁nús which was another word for "son", stemming from the root *sewH meaning "to give birth". This word has come down to us in English as "son". In the Romance languages however, both the words "son" and "daughter" are derived from the PIE source stemming from *dʰeh₁ (whence "daughter" comes), with Italian "figlio" and "figlia" coming ultimately (through Vulgar Latin) from the aforementioned PIE terms. So this is why "son" and "daughter" are nothing alike; the Germans basically said "nah bugger this, we prefer this word" and that's just how it is.

One down two to go. Why does "daughter" have a lot of silent letters? This is to do with the conservative nature of English spelling. See, at the time when the spelling standards for older variants of English (Middle English in this case) where being experimented with the spelling "daughter" didn't contain any silent letters. It would have been pronounced something like "dou-ch-ter" (with the "ch" here being my attempt to represent that funny guttural sound like in "loch" and somewhat like in German "ich"). If you can read IPA, this is /ˈdɔu̯xtər/. As time went on though English people got annoyed with having to make this unattractive guttural sound and grew bored of diphthongs (okay, that isn't exactly how it happened), and so what was once /ˈdɔu̯xtər/ became the pleasant sounding word we know today. That is, in spoken English it did. The spelling rules of English have not been properly updated for several centuries, and even then they were really rather archaic to say the least. This is why, for example, "night" and "knight" are pronounced the same today whereas, in Middle English, knight was pronounced /'kniçt/ (k-nicht -- "ch" being the sound in German "ich") and night /'niçt/. And this pretty much explains all the spelling irregularities and silent letters you see in English. French has much the same problem, perhaps worse. Languages like Finnish do not have this problem, the Finns being a sensible and hardy people who are far too busy shouting at bears and stopping Russians from stealing their land for potato cultivation to be worried about such silly things as "maintaining our linguistic heritage", far too rational to adhere to such thoughts as "purely phonetic spelling looks silly". Their language is pronounced exactly as it is written -- learn the sounds of the letters and you're practically fluent already!

Phew. Finally, onto question three. So you were correct in assuming that the ending -er (really this ending is -ter) has some etymological significance. You can probably already tell that the "-ter" at the end of "daughter" is a descendent of the *tḗr at the end of PIE *dʰugh₂tḗr. What you may not realise is that this "-ter" ending is etymologically from the same source as the "-ther" at the end of "brother", "mother" and "father", as well as the "-tor" at the end of "realtor", "gladiator" or "excavator". Words ending in "-tor" come to us in English through Latin, whereas words ending in "ther" come to us through Proto-Germanic, in which the sound "t" shifted to a "th" sound in most positions. Meaning that Latin "pater", "mater" are related to English "father", "mother" and both containing this ending derived from PIE *tḗr. *tḗr is an agentive suffix. The definition of this given by wiktionary is given as a suffix which:

"Derives agent nouns from verbs, denoting someone or something whose role or purpose it is to perform the verb's action."

Or in human speak, it tells you that its someone who does the thing. An excavator is something who excavates. A daughter is someone who suckles (remember the root *dʰeh meaning "to suckle". And thus, it stands to reason, a gladiator is someone who gladiates.^1 The fascinating thing about language is that nothing is arbitrary -- people don't really just decide to randomly start adding sounds onto words for no reason. If it's there, then there's almost certainly a reason for it and that reason can be uncovered and traced back with the comparative method and that's what historical linguistics is all about.

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^1 Gladius. It's someone who wields a gladius (just the Latin word for any old sword).