It's only speculation, but it seems very likely to me that the Targaryens are (at least partially) inspired by Vikings. My arguments start out rather vague, but hear me out!
The names may be inspired by the Viking gods, the Aesir/Æsir/Aser. As one comment mentioned, ae/æ is indeed a single letter in Danish, and as far as I know, the sound of it entered English around the time when Denmark ruled England, during which time Danish (or "Old Norse") had a major impact on English, including syntax changes and a huge amount of nouns entering the language, mostly related to travel, working and living. In English, the a and e tend to be split up on either side of a consonant, such as in words like cave, make, or fake. While those particular words do not derive from Danish as far as I know ("cake" does, though), the ae sound is exactly the same as the æ in Danish and, to my knowledge, comes from there.
George R. R. Martin certainly did a lot of research in European medieval history and surely must've read a bit about the Aesir as well as other European gods as inspiration for magic in his world. In fact, the Targaryens' relationship to dragons is very likely to come from Vikings as well. Vikings are most famous for their ties to dragons through their ships, whose ends had carved dragon heads, leading to the word dragon being a synonym for ship/boat and later sail. Monks surviving the attack on Lindisfarne, one of the earliest Viking raids, wrote stories about the attack which had a huge impact on how all of Europe would view the Vikings as heartless brutes in the future, even though the monks may have been exaggerating their stories a bit - indeed those stories claim that the viking raiders were commanding DRAGONS! Not just the ships, but the real, flying, fire-breathing ones. (Yeah right, monks)
Recap: They have dragons and their gods are the Aesir. Also, they are notorious for their blonde hair. I rest my case.